I am done

rl/Lib/site-packages/sympy/functions/special/__init__.py

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+# Stub __init__.py for the sympy.functions.special package

rl/Lib/site-packages/sympy/functions/special/benchmarks/bench_special.py

@@ -0,0 +1,8 @@
+from sympy.core.symbol import symbols
+from sympy.functions.special.spherical_harmonics import Ynm
+
+x, y = symbols('x,y')
+
+
+def timeit_Ynm_xy():
+    Ynm(1, 1, x, y)

rl/Lib/site-packages/sympy/functions/special/beta_functions.py

@@ -0,0 +1,389 @@
+from sympy.core import S
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.core.symbol import Dummy, uniquely_named_symbol
+from sympy.functions.special.gamma_functions import gamma, digamma
+from sympy.functions.combinatorial.numbers import catalan
+from sympy.functions.elementary.complexes import conjugate
+
+# See mpmath #569 and SymPy #20569
+def betainc_mpmath_fix(a, b, x1, x2, reg=0):
+    from mpmath import betainc, mpf
+    if x1 == x2:
+        return mpf(0)
+    else:
+        return betainc(a, b, x1, x2, reg)
+
+###############################################################################
+############################ COMPLETE BETA  FUNCTION ##########################
+###############################################################################
+
+class beta(Function):
+    r"""
+    The beta integral is called the Eulerian integral of the first kind by
+    Legendre:
+
+    .. math::
+        \mathrm{B}(x,y)  \int^{1}_{0} t^{x-1} (1-t)^{y-1} \mathrm{d}t.
+
+    Explanation
+    ===========
+
+    The Beta function or Euler's first integral is closely associated
+    with the gamma function. The Beta function is often used in probability
+    theory and mathematical statistics. It satisfies properties like:
+
+    .. math::
+        \mathrm{B}(a,1) = \frac{1}{a} \\
+        \mathrm{B}(a,b) = \mathrm{B}(b,a)  \\
+        \mathrm{B}(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}
+
+    Therefore for integral values of $a$ and $b$:
+
+    .. math::
+        \mathrm{B} = \frac{(a-1)! (b-1)!}{(a+b-1)!}
+
+    A special case of the Beta function when `x = y` is the
+    Central Beta function. It satisfies properties like:
+
+    .. math::
+        \mathrm{B}(x) = 2^{1 - 2x}\mathrm{B}(x, \frac{1}{2})
+        \mathrm{B}(x) = 2^{1 - 2x} cos(\pi x) \mathrm{B}(\frac{1}{2} - x, x)
+        \mathrm{B}(x) = \int_{0}^{1} \frac{t^x}{(1 + t)^{2x}} dt
+        \mathrm{B}(x) = \frac{2}{x} \prod_{n = 1}^{\infty} \frac{n(n + 2x)}{(n + x)^2}
+
+    Examples
+    ========
+
+    >>> from sympy import I, pi
+    >>> from sympy.abc import x, y
+
+    The Beta function obeys the mirror symmetry:
+
+    >>> from sympy import beta, conjugate
+    >>> conjugate(beta(x, y))
+    beta(conjugate(x), conjugate(y))
+
+    Differentiation with respect to both $x$ and $y$ is supported:
+
+    >>> from sympy import beta, diff
+    >>> diff(beta(x, y), x)
+    (polygamma(0, x) - polygamma(0, x + y))*beta(x, y)
+
+    >>> diff(beta(x, y), y)
+    (polygamma(0, y) - polygamma(0, x + y))*beta(x, y)
+
+    >>> diff(beta(x), x)
+    2*(polygamma(0, x) - polygamma(0, 2*x))*beta(x, x)
+
+    We can numerically evaluate the Beta function to
+    arbitrary precision for any complex numbers x and y:
+
+    >>> from sympy import beta
+    >>> beta(pi).evalf(40)
+    0.02671848900111377452242355235388489324562
+
+    >>> beta(1 + I).evalf(20)
+    -0.2112723729365330143 - 0.7655283165378005676*I
+
+    See Also
+    ========
+
+    gamma: Gamma function.
+    uppergamma: Upper incomplete gamma function.
+    lowergamma: Lower incomplete gamma function.
+    polygamma: Polygamma function.
+    loggamma: Log Gamma function.
+    digamma: Digamma function.
+    trigamma: Trigamma function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Beta_function
+    .. [2] https://mathworld.wolfram.com/BetaFunction.html
+    .. [3] https://dlmf.nist.gov/5.12
+
+    """
+    unbranched = True
+
+    def fdiff(self, argindex):
+        x, y = self.args
+        if argindex == 1:
+            # Diff wrt x
+            return beta(x, y)*(digamma(x) - digamma(x + y))
+        elif argindex == 2:
+            # Diff wrt y
+            return beta(x, y)*(digamma(y) - digamma(x + y))
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, x, y=None):
+        if y is None:
+            return beta(x, x)
+        if x.is_Number and y.is_Number:
+            return beta(x, y, evaluate=False).doit()
+
+    def doit(self, **hints):
+        x = xold = self.args[0]
+        # Deal with unevaluated single argument beta
+        single_argument = len(self.args) == 1
+        y = yold = self.args[0] if single_argument else self.args[1]
+        if hints.get('deep', True):
+            x = x.doit(**hints)
+            y = y.doit(**hints)
+        if y.is_zero or x.is_zero:
+            return S.ComplexInfinity
+        if y is S.One:
+            return 1/x
+        if x is S.One:
+            return 1/y
+        if y == x + 1:
+            return 1/(x*y*catalan(x))
+        s = x + y
+        if (s.is_integer and s.is_negative and x.is_integer is False and
+            y.is_integer is False):
+            return S.Zero
+        if x == xold and y == yold and not single_argument:
+            return self
+        return beta(x, y)
+
+    def _eval_expand_func(self, **hints):
+        x, y = self.args
+        return gamma(x)*gamma(y) / gamma(x + y)
+
+    def _eval_is_real(self):
+        return self.args[0].is_real and self.args[1].is_real
+
+    def _eval_conjugate(self):
+        return self.func(self.args[0].conjugate(), self.args[1].conjugate())
+
+    def _eval_rewrite_as_gamma(self, x, y, piecewise=True, **kwargs):
+        return self._eval_expand_func(**kwargs)
+
+    def _eval_rewrite_as_Integral(self, x, y, **kwargs):
+        from sympy.integrals.integrals import Integral
+        t = Dummy(uniquely_named_symbol('t', [x, y]).name)
+        return Integral(t**(x - 1)*(1 - t)**(y - 1), (t, 0, 1))
+
+###############################################################################
+########################## INCOMPLETE BETA FUNCTION ###########################
+###############################################################################
+
+class betainc(Function):
+    r"""
+    The Generalized Incomplete Beta function is defined as
+
+    .. math::
+        \mathrm{B}_{(x_1, x_2)}(a, b) = \int_{x_1}^{x_2} t^{a - 1} (1 - t)^{b - 1} dt
+
+    The Incomplete Beta function is a special case
+    of the Generalized Incomplete Beta function :
+
+    .. math:: \mathrm{B}_z (a, b) = \mathrm{B}_{(0, z)}(a, b)
+
+    The Incomplete Beta function satisfies :
+
+    .. math:: \mathrm{B}_z (a, b) = (-1)^a \mathrm{B}_{\frac{z}{z - 1}} (a, 1 - a - b)
+
+    The Beta function is a special case of the Incomplete Beta function :
+
+    .. math:: \mathrm{B}(a, b) = \mathrm{B}_{1}(a, b)
+
+    Examples
+    ========
+
+    >>> from sympy import betainc, symbols, conjugate
+    >>> a, b, x, x1, x2 = symbols('a b x x1 x2')
+
+    The Generalized Incomplete Beta function is given by:
+
+    >>> betainc(a, b, x1, x2)
+    betainc(a, b, x1, x2)
+
+    The Incomplete Beta function can be obtained as follows:
+
+    >>> betainc(a, b, 0, x)
+    betainc(a, b, 0, x)
+
+    The Incomplete Beta function obeys the mirror symmetry:
+
+    >>> conjugate(betainc(a, b, x1, x2))
+    betainc(conjugate(a), conjugate(b), conjugate(x1), conjugate(x2))
+
+    We can numerically evaluate the Incomplete Beta function to
+    arbitrary precision for any complex numbers a, b, x1 and x2:
+
+    >>> from sympy import betainc, I
+    >>> betainc(2, 3, 4, 5).evalf(10)
+    56.08333333
+    >>> betainc(0.75, 1 - 4*I, 0, 2 + 3*I).evalf(25)
+    0.2241657956955709603655887 + 0.3619619242700451992411724*I
+
+    The Generalized Incomplete Beta function can be expressed
+    in terms of the Generalized Hypergeometric function.
+
+    >>> from sympy import hyper
+    >>> betainc(a, b, x1, x2).rewrite(hyper)
+    (-x1**a*hyper((a, 1 - b), (a + 1,), x1) + x2**a*hyper((a, 1 - b), (a + 1,), x2))/a
+
+    See Also
+    ========
+
+    beta: Beta function
+    hyper: Generalized Hypergeometric function
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function
+    .. [2] https://dlmf.nist.gov/8.17
+    .. [3] https://functions.wolfram.com/GammaBetaErf/Beta4/
+    .. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/
+
+    """
+    nargs = 4
+    unbranched = True
+
+    def fdiff(self, argindex):
+        a, b, x1, x2 = self.args
+        if argindex == 3:
+            # Diff wrt x1
+            return -(1 - x1)**(b - 1)*x1**(a - 1)
+        elif argindex == 4:
+            # Diff wrt x2
+            return (1 - x2)**(b - 1)*x2**(a - 1)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_mpmath(self):
+        return betainc_mpmath_fix, self.args
+
+    def _eval_is_real(self):
+        if all(arg.is_real for arg in self.args):
+            return True
+
+    def _eval_conjugate(self):
+        return self.func(*map(conjugate, self.args))
+
+    def _eval_rewrite_as_Integral(self, a, b, x1, x2, **kwargs):
+        from sympy.integrals.integrals import Integral
+        t = Dummy(uniquely_named_symbol('t', [a, b, x1, x2]).name)
+        return Integral(t**(a - 1)*(1 - t)**(b - 1), (t, x1, x2))
+
+    def _eval_rewrite_as_hyper(self, a, b, x1, x2, **kwargs):
+        from sympy.functions.special.hyper import hyper
+        return (x2**a * hyper((a, 1 - b), (a + 1,), x2) - x1**a * hyper((a, 1 - b), (a + 1,), x1)) / a
+
+###############################################################################
+#################### REGULARIZED INCOMPLETE BETA FUNCTION #####################
+###############################################################################
+
+class betainc_regularized(Function):
+    r"""
+    The Generalized Regularized Incomplete Beta function is given by
+
+    .. math::
+        \mathrm{I}_{(x_1, x_2)}(a, b) = \frac{\mathrm{B}_{(x_1, x_2)}(a, b)}{\mathrm{B}(a, b)}
+
+    The Regularized Incomplete Beta function is a special case
+    of the Generalized Regularized Incomplete Beta function :
+
+    .. math:: \mathrm{I}_z (a, b) = \mathrm{I}_{(0, z)}(a, b)
+
+    The Regularized Incomplete Beta function is the cumulative distribution
+    function of the beta distribution.
+
+    Examples
+    ========
+
+    >>> from sympy import betainc_regularized, symbols, conjugate
+    >>> a, b, x, x1, x2 = symbols('a b x x1 x2')
+
+    The Generalized Regularized Incomplete Beta
+    function is given by:
+
+    >>> betainc_regularized(a, b, x1, x2)
+    betainc_regularized(a, b, x1, x2)
+
+    The Regularized Incomplete Beta function
+    can be obtained as follows:
+
+    >>> betainc_regularized(a, b, 0, x)
+    betainc_regularized(a, b, 0, x)
+
+    The Regularized Incomplete Beta function
+    obeys the mirror symmetry:
+
+    >>> conjugate(betainc_regularized(a, b, x1, x2))
+    betainc_regularized(conjugate(a), conjugate(b), conjugate(x1), conjugate(x2))
+
+    We can numerically evaluate the Regularized Incomplete Beta function
+    to arbitrary precision for any complex numbers a, b, x1 and x2:
+
+    >>> from sympy import betainc_regularized, pi, E
+    >>> betainc_regularized(1, 2, 0, 0.25).evalf(10)
+    0.4375000000
+    >>> betainc_regularized(pi, E, 0, 1).evalf(5)
+    1.00000
+
+    The Generalized Regularized Incomplete Beta function can be
+    expressed in terms of the Generalized Hypergeometric function.
+
+    >>> from sympy import hyper
+    >>> betainc_regularized(a, b, x1, x2).rewrite(hyper)
+    (-x1**a*hyper((a, 1 - b), (a + 1,), x1) + x2**a*hyper((a, 1 - b), (a + 1,), x2))/(a*beta(a, b))
+
+    See Also
+    ========
+
+    beta: Beta function
+    hyper: Generalized Hypergeometric function
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function
+    .. [2] https://dlmf.nist.gov/8.17
+    .. [3] https://functions.wolfram.com/GammaBetaErf/Beta4/
+    .. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/
+
+    """
+    nargs = 4
+    unbranched = True
+
+    def __new__(cls, a, b, x1, x2):
+        return Function.__new__(cls, a, b, x1, x2)
+
+    def _eval_mpmath(self):
+        return betainc_mpmath_fix, (*self.args, S(1))
+
+    def fdiff(self, argindex):
+        a, b, x1, x2 = self.args
+        if argindex == 3:
+            # Diff wrt x1
+            return -(1 - x1)**(b - 1)*x1**(a - 1) / beta(a, b)
+        elif argindex == 4:
+            # Diff wrt x2
+            return (1 - x2)**(b - 1)*x2**(a - 1) / beta(a, b)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_is_real(self):
+        if all(arg.is_real for arg in self.args):
+            return True
+
+    def _eval_conjugate(self):
+        return self.func(*map(conjugate, self.args))
+
+    def _eval_rewrite_as_Integral(self, a, b, x1, x2, **kwargs):
+        from sympy.integrals.integrals import Integral
+        t = Dummy(uniquely_named_symbol('t', [a, b, x1, x2]).name)
+        integrand = t**(a - 1)*(1 - t)**(b - 1)
+        expr = Integral(integrand, (t, x1, x2))
+        return expr / Integral(integrand, (t, 0, 1))
+
+    def _eval_rewrite_as_hyper(self, a, b, x1, x2, **kwargs):
+        from sympy.functions.special.hyper import hyper
+        expr = (x2**a * hyper((a, 1 - b), (a + 1,), x2) - x1**a * hyper((a, 1 - b), (a + 1,), x1)) / a
+        return expr / beta(a, b)

rl/Lib/site-packages/sympy/functions/special/bsplines.py

@@ -0,0 +1,351 @@
+from sympy.core import S, sympify
+from sympy.core.symbol import (Dummy, symbols)
+from sympy.functions import Piecewise, piecewise_fold
+from sympy.logic.boolalg import And
+from sympy.sets.sets import Interval
+
+from functools import lru_cache
+
+
+def _ivl(cond, x):
+    """return the interval corresponding to the condition
+
+    Conditions in spline's Piecewise give the range over
+    which an expression is valid like (lo <= x) & (x <= hi).
+    This function returns (lo, hi).
+    """
+    if isinstance(cond, And) and len(cond.args) == 2:
+        a, b = cond.args
+        if a.lts == x:
+            a, b = b, a
+        return a.lts, b.gts
+    raise TypeError('unexpected cond type: %s' % cond)
+
+
+def _add_splines(c, b1, d, b2, x):
+    """Construct c*b1 + d*b2."""
+
+    if S.Zero in (b1, c):
+        rv = piecewise_fold(d * b2)
+    elif S.Zero in (b2, d):
+        rv = piecewise_fold(c * b1)
+    else:
+        new_args = []
+        # Just combining the Piecewise without any fancy optimization
+        p1 = piecewise_fold(c * b1)
+        p2 = piecewise_fold(d * b2)
+
+        # Search all Piecewise arguments except (0, True)
+        p2args = list(p2.args[:-1])
+
+        # This merging algorithm assumes the conditions in
+        # p1 and p2 are sorted
+        for arg in p1.args[:-1]:
+            expr = arg.expr
+            cond = arg.cond
+
+            lower = _ivl(cond, x)[0]
+
+            # Check p2 for matching conditions that can be merged
+            for i, arg2 in enumerate(p2args):
+                expr2 = arg2.expr
+                cond2 = arg2.cond
+
+                lower_2, upper_2 = _ivl(cond2, x)
+                if cond2 == cond:
+                    # Conditions match, join expressions
+                    expr += expr2
+                    # Remove matching element
+                    del p2args[i]
+                    # No need to check the rest
+                    break
+                elif lower_2 < lower and upper_2 <= lower:
+                    # Check if arg2 condition smaller than arg1,
+                    # add to new_args by itself (no match expected
+                    # in p1)
+                    new_args.append(arg2)
+                    del p2args[i]
+                    break
+
+            # Checked all, add expr and cond
+            new_args.append((expr, cond))
+
+        # Add remaining items from p2args
+        new_args.extend(p2args)
+
+        # Add final (0, True)
+        new_args.append((0, True))
+
+        rv = Piecewise(*new_args, evaluate=False)
+
+    return rv.expand()
+
+
+@lru_cache(maxsize=128)
+def bspline_basis(d, knots, n, x):
+    """
+    The $n$-th B-spline at $x$ of degree $d$ with knots.
+
+    Explanation
+    ===========
+
+    B-Splines are piecewise polynomials of degree $d$. They are defined on a
+    set of knots, which is a sequence of integers or floats.
+
+    Examples
+    ========
+
+    The 0th degree splines have a value of 1 on a single interval:
+
+        >>> from sympy import bspline_basis
+        >>> from sympy.abc import x
+        >>> d = 0
+        >>> knots = tuple(range(5))
+        >>> bspline_basis(d, knots, 0, x)
+        Piecewise((1, (x >= 0) & (x <= 1)), (0, True))
+
+    For a given ``(d, knots)`` there are ``len(knots)-d-1`` B-splines
+    defined, that are indexed by ``n`` (starting at 0).
+
+    Here is an example of a cubic B-spline:
+
+        >>> bspline_basis(3, tuple(range(5)), 0, x)
+        Piecewise((x**3/6, (x >= 0) & (x <= 1)),
+                  (-x**3/2 + 2*x**2 - 2*x + 2/3,
+                  (x >= 1) & (x <= 2)),
+                  (x**3/2 - 4*x**2 + 10*x - 22/3,
+                  (x >= 2) & (x <= 3)),
+                  (-x**3/6 + 2*x**2 - 8*x + 32/3,
+                  (x >= 3) & (x <= 4)),
+                  (0, True))
+
+    By repeating knot points, you can introduce discontinuities in the
+    B-splines and their derivatives:
+
+        >>> d = 1
+        >>> knots = (0, 0, 2, 3, 4)
+        >>> bspline_basis(d, knots, 0, x)
+        Piecewise((1 - x/2, (x >= 0) & (x <= 2)), (0, True))
+
+    It is quite time consuming to construct and evaluate B-splines. If
+    you need to evaluate a B-spline many times, it is best to lambdify them
+    first:
+
+        >>> from sympy import lambdify
+        >>> d = 3
+        >>> knots = tuple(range(10))
+        >>> b0 = bspline_basis(d, knots, 0, x)
+        >>> f = lambdify(x, b0)
+        >>> y = f(0.5)
+
+    Parameters
+    ==========
+
+    d : integer
+        degree of bspline
+
+    knots : list of integer values
+        list of knots points of bspline
+
+    n : integer
+        $n$-th B-spline
+
+    x : symbol
+
+    See Also
+    ========
+
+    bspline_basis_set
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/B-spline
+
+    """
+    # make sure x has no assumptions so conditions don't evaluate
+    xvar = x
+    x = Dummy()
+
+    knots = tuple(sympify(k) for k in knots)
+    d = int(d)
+    n = int(n)
+    n_knots = len(knots)
+    n_intervals = n_knots - 1
+    if n + d + 1 > n_intervals:
+        raise ValueError("n + d + 1 must not exceed len(knots) - 1")
+    if d == 0:
+        result = Piecewise(
+            (S.One, Interval(knots[n], knots[n + 1]).contains(x)), (0, True)
+        )
+    elif d > 0:
+        denom = knots[n + d + 1] - knots[n + 1]
+        if denom != S.Zero:
+            B = (knots[n + d + 1] - x) / denom
+            b2 = bspline_basis(d - 1, knots, n + 1, x)
+        else:
+            b2 = B = S.Zero
+
+        denom = knots[n + d] - knots[n]
+        if denom != S.Zero:
+            A = (x - knots[n]) / denom
+            b1 = bspline_basis(d - 1, knots, n, x)
+        else:
+            b1 = A = S.Zero
+
+        result = _add_splines(A, b1, B, b2, x)
+    else:
+        raise ValueError("degree must be non-negative: %r" % n)
+
+    # return result with user-given x
+    return result.xreplace({x: xvar})
+
+
+def bspline_basis_set(d, knots, x):
+    """
+    Return the ``len(knots)-d-1`` B-splines at *x* of degree *d*
+    with *knots*.
+
+    Explanation
+    ===========
+
+    This function returns a list of piecewise polynomials that are the
+    ``len(knots)-d-1`` B-splines of degree *d* for the given knots.
+    This function calls ``bspline_basis(d, knots, n, x)`` for different
+    values of *n*.
+
+    Examples
+    ========
+
+    >>> from sympy import bspline_basis_set
+    >>> from sympy.abc import x
+    >>> d = 2
+    >>> knots = range(5)
+    >>> splines = bspline_basis_set(d, knots, x)
+    >>> splines
+    [Piecewise((x**2/2, (x >= 0) & (x <= 1)),
+               (-x**2 + 3*x - 3/2, (x >= 1) & (x <= 2)),
+               (x**2/2 - 3*x + 9/2, (x >= 2) & (x <= 3)),
+               (0, True)),
+    Piecewise((x**2/2 - x + 1/2, (x >= 1) & (x <= 2)),
+              (-x**2 + 5*x - 11/2, (x >= 2) & (x <= 3)),
+              (x**2/2 - 4*x + 8, (x >= 3) & (x <= 4)),
+              (0, True))]
+
+    Parameters
+    ==========
+
+    d : integer
+        degree of bspline
+
+    knots : list of integers
+        list of knots points of bspline
+
+    x : symbol
+
+    See Also
+    ========
+
+    bspline_basis
+
+    """
+    n_splines = len(knots) - d - 1
+    return [bspline_basis(d, tuple(knots), i, x) for i in range(n_splines)]
+
+
+def interpolating_spline(d, x, X, Y):
+    """
+    Return spline of degree *d*, passing through the given *X*
+    and *Y* values.
+
+    Explanation
+    ===========
+
+    This function returns a piecewise function such that each part is
+    a polynomial of degree not greater than *d*. The value of *d*
+    must be 1 or greater and the values of *X* must be strictly
+    increasing.
+
+    Examples
+    ========
+
+    >>> from sympy import interpolating_spline
+    >>> from sympy.abc import x
+    >>> interpolating_spline(1, x, [1, 2, 4, 7], [3, 6, 5, 7])
+    Piecewise((3*x, (x >= 1) & (x <= 2)),
+            (7 - x/2, (x >= 2) & (x <= 4)),
+            (2*x/3 + 7/3, (x >= 4) & (x <= 7)))
+    >>> interpolating_spline(3, x, [-2, 0, 1, 3, 4], [4, 2, 1, 1, 3])
+    Piecewise((7*x**3/117 + 7*x**2/117 - 131*x/117 + 2, (x >= -2) & (x <= 1)),
+            (10*x**3/117 - 2*x**2/117 - 122*x/117 + 77/39, (x >= 1) & (x <= 4)))
+
+    Parameters
+    ==========
+
+    d : integer
+        Degree of Bspline strictly greater than equal to one
+
+    x : symbol
+
+    X : list of strictly increasing real values
+        list of X coordinates through which the spline passes
+
+    Y : list of real values
+        list of corresponding Y coordinates through which the spline passes
+
+    See Also
+    ========
+
+    bspline_basis_set, interpolating_poly
+
+    """
+    from sympy.solvers.solveset import linsolve
+    from sympy.matrices.dense import Matrix
+
+    # Input sanitization
+    d = sympify(d)
+    if not (d.is_Integer and d.is_positive):
+        raise ValueError("Spline degree must be a positive integer, not %s." % d)
+    if len(X) != len(Y):
+        raise ValueError("Number of X and Y coordinates must be the same.")
+    if len(X) < d + 1:
+        raise ValueError("Degree must be less than the number of control points.")
+    if not all(a < b for a, b in zip(X, X[1:])):
+        raise ValueError("The x-coordinates must be strictly increasing.")
+    X = [sympify(i) for i in X]
+
+    # Evaluating knots value
+    if d.is_odd:
+        j = (d + 1) // 2
+        interior_knots = X[j:-j]
+    else:
+        j = d // 2
+        interior_knots = [
+            (a + b)/2 for a, b in zip(X[j : -j - 1], X[j + 1 : -j])
+        ]
+
+    knots = [X[0]] * (d + 1) + list(interior_knots) + [X[-1]] * (d + 1)
+
+    basis = bspline_basis_set(d, knots, x)
+
+    A = [[b.subs(x, v) for b in basis] for v in X]
+
+    coeff = linsolve((Matrix(A), Matrix(Y)), symbols("c0:{}".format(len(X)), cls=Dummy))
+    coeff = list(coeff)[0]
+    intervals = {c for b in basis for (e, c) in b.args if c != True}
+
+    # Sorting the intervals
+    #  ival contains the end-points of each interval
+    ival = [_ivl(c, x) for c in intervals]
+    com = zip(ival, intervals)
+    com = sorted(com, key=lambda x: x[0])
+    intervals = [y for x, y in com]
+
+    basis_dicts = [{c: e for (e, c) in b.args} for b in basis]
+    spline = []
+    for i in intervals:
+        piece = sum(
+            [c * d.get(i, S.Zero) for (c, d) in zip(coeff, basis_dicts)], S.Zero
+        )
+        spline.append((piece, i))
+    return Piecewise(*spline)

rl/Lib/site-packages/sympy/functions/special/delta_functions.py

@@ -0,0 +1,664 @@
+from sympy.core import S, diff
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.core.logic import fuzzy_not
+from sympy.core.relational import Eq, Ne
+from sympy.functions.elementary.complexes import im, sign
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.polys.polyerrors import PolynomialError
+from sympy.polys.polyroots import roots
+from sympy.utilities.misc import filldedent
+
+
+###############################################################################
+################################ DELTA FUNCTION ###############################
+###############################################################################
+
+
+class DiracDelta(Function):
+    r"""
+    The DiracDelta function and its derivatives.
+
+    Explanation
+    ===========
+
+    DiracDelta is not an ordinary function. It can be rigorously defined either
+    as a distribution or as a measure.
+
+    DiracDelta only makes sense in definite integrals, and in particular,
+    integrals of the form ``Integral(f(x)*DiracDelta(x - x0), (x, a, b))``,
+    where it equals ``f(x0)`` if ``a <= x0 <= b`` and ``0`` otherwise. Formally,
+    DiracDelta acts in some ways like a function that is ``0`` everywhere except
+    at ``0``, but in many ways it also does not. It can often be useful to treat
+    DiracDelta in formal ways, building up and manipulating expressions with
+    delta functions (which may eventually be integrated), but care must be taken
+    to not treat it as a real function. SymPy's ``oo`` is similar. It only
+    truly makes sense formally in certain contexts (such as integration limits),
+    but SymPy allows its use everywhere, and it tries to be consistent with
+    operations on it (like ``1/oo``), but it is easy to get into trouble and get
+    wrong results if ``oo`` is treated too much like a number. Similarly, if
+    DiracDelta is treated too much like a function, it is easy to get wrong or
+    nonsensical results.
+
+    DiracDelta function has the following properties:
+
+    1) $\frac{d}{d x} \theta(x) = \delta(x)$
+    2) $\int_{-\infty}^\infty \delta(x - a)f(x)\, dx = f(a)$ and $\int_{a-
+       \epsilon}^{a+\epsilon} \delta(x - a)f(x)\, dx = f(a)$
+    3) $\delta(x) = 0$ for all $x \neq 0$
+    4) $\delta(g(x)) = \sum_i \frac{\delta(x - x_i)}{\|g'(x_i)\|}$ where $x_i$
+       are the roots of $g$
+    5) $\delta(-x) = \delta(x)$
+
+    Derivatives of ``k``-th order of DiracDelta have the following properties:
+
+    6) $\delta(x, k) = 0$ for all $x \neq 0$
+    7) $\delta(-x, k) = -\delta(x, k)$ for odd $k$
+    8) $\delta(-x, k) = \delta(x, k)$ for even $k$
+
+    Examples
+    ========
+
+    >>> from sympy import DiracDelta, diff, pi
+    >>> from sympy.abc import x, y
+
+    >>> DiracDelta(x)
+    DiracDelta(x)
+    >>> DiracDelta(1)
+    0
+    >>> DiracDelta(-1)
+    0
+    >>> DiracDelta(pi)
+    0
+    >>> DiracDelta(x - 4).subs(x, 4)
+    DiracDelta(0)
+    >>> diff(DiracDelta(x))
+    DiracDelta(x, 1)
+    >>> diff(DiracDelta(x - 1), x, 2)
+    DiracDelta(x - 1, 2)
+    >>> diff(DiracDelta(x**2 - 1), x, 2)
+    2*(2*x**2*DiracDelta(x**2 - 1, 2) + DiracDelta(x**2 - 1, 1))
+    >>> DiracDelta(3*x).is_simple(x)
+    True
+    >>> DiracDelta(x**2).is_simple(x)
+    False
+    >>> DiracDelta((x**2 - 1)*y).expand(diracdelta=True, wrt=x)
+    DiracDelta(x - 1)/(2*Abs(y)) + DiracDelta(x + 1)/(2*Abs(y))
+
+    See Also
+    ========
+
+    Heaviside
+    sympy.simplify.simplify.simplify, is_simple
+    sympy.functions.special.tensor_functions.KroneckerDelta
+
+    References
+    ==========
+
+    .. [1] https://mathworld.wolfram.com/DeltaFunction.html
+
+    """
+
+    is_real = True
+
+    def fdiff(self, argindex=1):
+        """
+        Returns the first derivative of a DiracDelta Function.
+
+        Explanation
+        ===========
+
+        The difference between ``diff()`` and ``fdiff()`` is: ``diff()`` is the
+        user-level function and ``fdiff()`` is an object method. ``fdiff()`` is
+        a convenience method available in the ``Function`` class. It returns
+        the derivative of the function without considering the chain rule.
+        ``diff(function, x)`` calls ``Function._eval_derivative`` which in turn
+        calls ``fdiff()`` internally to compute the derivative of the function.
+
+        Examples
+        ========
+
+        >>> from sympy import DiracDelta, diff
+        >>> from sympy.abc import x
+
+        >>> DiracDelta(x).fdiff()
+        DiracDelta(x, 1)
+
+        >>> DiracDelta(x, 1).fdiff()
+        DiracDelta(x, 2)
+
+        >>> DiracDelta(x**2 - 1).fdiff()
+        DiracDelta(x**2 - 1, 1)
+
+        >>> diff(DiracDelta(x, 1)).fdiff()
+        DiracDelta(x, 3)
+
+        Parameters
+        ==========
+
+        argindex : integer
+            degree of derivative
+
+        """
+        if argindex == 1:
+            #I didn't know if there is a better way to handle default arguments
+            k = 0
+            if len(self.args) > 1:
+                k = self.args[1]
+            return self.func(self.args[0], k + 1)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, arg, k=S.Zero):
+        """
+        Returns a simplified form or a value of DiracDelta depending on the
+        argument passed by the DiracDelta object.
+
+        Explanation
+        ===========
+
+        The ``eval()`` method is automatically called when the ``DiracDelta``
+        class is about to be instantiated and it returns either some simplified
+        instance or the unevaluated instance depending on the argument passed.
+        In other words, ``eval()`` method is not needed to be called explicitly,
+        it is being called and evaluated once the object is called.
+
+        Examples
+        ========
+
+        >>> from sympy import DiracDelta, S
+        >>> from sympy.abc import x
+
+        >>> DiracDelta(x)
+        DiracDelta(x)
+
+        >>> DiracDelta(-x, 1)
+        -DiracDelta(x, 1)
+
+        >>> DiracDelta(1)
+        0
+
+        >>> DiracDelta(5, 1)
+        0
+
+        >>> DiracDelta(0)
+        DiracDelta(0)
+
+        >>> DiracDelta(-1)
+        0
+
+        >>> DiracDelta(S.NaN)
+        nan
+
+        >>> DiracDelta(x - 100).subs(x, 5)
+        0
+
+        >>> DiracDelta(x - 100).subs(x, 100)
+        DiracDelta(0)
+
+        Parameters
+        ==========
+
+        k : integer
+            order of derivative
+
+        arg : argument passed to DiracDelta
+
+        """
+        if not k.is_Integer or k.is_negative:
+            raise ValueError("Error: the second argument of DiracDelta must be \
+            a non-negative integer, %s given instead." % (k,))
+        if arg is S.NaN:
+            return S.NaN
+        if arg.is_nonzero:
+            return S.Zero
+        if fuzzy_not(im(arg).is_zero):
+            raise ValueError(filldedent('''
+                Function defined only for Real Values.
+                Complex part: %s  found in %s .''' % (
+                repr(im(arg)), repr(arg))))
+        c, nc = arg.args_cnc()
+        if c and c[0] is S.NegativeOne:
+            # keep this fast and simple instead of using
+            # could_extract_minus_sign
+            if k.is_odd:
+                return -cls(-arg, k)
+            elif k.is_even:
+                return cls(-arg, k) if k else cls(-arg)
+        elif k.is_zero:
+            return cls(arg, evaluate=False)
+
+    def _eval_expand_diracdelta(self, **hints):
+        """
+        Compute a simplified representation of the function using
+        property number 4. Pass ``wrt`` as a hint to expand the expression
+        with respect to a particular variable.
+
+        Explanation
+        ===========
+
+        ``wrt`` is:
+
+        - a variable with respect to which a DiracDelta expression will
+        get expanded.
+
+        Examples
+        ========
+
+        >>> from sympy import DiracDelta
+        >>> from sympy.abc import x, y
+
+        >>> DiracDelta(x*y).expand(diracdelta=True, wrt=x)
+        DiracDelta(x)/Abs(y)
+        >>> DiracDelta(x*y).expand(diracdelta=True, wrt=y)
+        DiracDelta(y)/Abs(x)
+
+        >>> DiracDelta(x**2 + x - 2).expand(diracdelta=True, wrt=x)
+        DiracDelta(x - 1)/3 + DiracDelta(x + 2)/3
+
+        See Also
+        ========
+
+        is_simple, Diracdelta
+
+        """
+        wrt = hints.get('wrt', None)
+        if wrt is None:
+            free = self.free_symbols
+            if len(free) == 1:
+                wrt = free.pop()
+            else:
+                raise TypeError(filldedent('''
+            When there is more than 1 free symbol or variable in the expression,
+            the 'wrt' keyword is required as a hint to expand when using the
+            DiracDelta hint.'''))
+
+        if not self.args[0].has(wrt) or (len(self.args) > 1 and self.args[1] != 0 ):
+            return self
+        try:
+            argroots = roots(self.args[0], wrt)
+            result = 0
+            valid = True
+            darg = abs(diff(self.args[0], wrt))
+            for r, m in argroots.items():
+                if r.is_real is not False and m == 1:
+                    result += self.func(wrt - r)/darg.subs(wrt, r)
+                else:
+                    # don't handle non-real and if m != 1 then
+                    # a polynomial will have a zero in the derivative (darg)
+                    # at r
+                    valid = False
+                    break
+            if valid:
+                return result
+        except PolynomialError:
+            pass
+        return self
+
+    def is_simple(self, x):
+        """
+        Tells whether the argument(args[0]) of DiracDelta is a linear
+        expression in *x*.
+
+        Examples
+        ========
+
+        >>> from sympy import DiracDelta, cos
+        >>> from sympy.abc import x, y
+
+        >>> DiracDelta(x*y).is_simple(x)
+        True
+        >>> DiracDelta(x*y).is_simple(y)
+        True
+
+        >>> DiracDelta(x**2 + x - 2).is_simple(x)
+        False
+
+        >>> DiracDelta(cos(x)).is_simple(x)
+        False
+
+        Parameters
+        ==========
+
+        x : can be a symbol
+
+        See Also
+        ========
+
+        sympy.simplify.simplify.simplify, DiracDelta
+
+        """
+        p = self.args[0].as_poly(x)
+        if p:
+            return p.degree() == 1
+        return False
+
+    def _eval_rewrite_as_Piecewise(self, *args, **kwargs):
+        """
+        Represents DiracDelta in a piecewise form.
+
+        Examples
+        ========
+
+        >>> from sympy import DiracDelta, Piecewise, Symbol
+        >>> x = Symbol('x')
+
+        >>> DiracDelta(x).rewrite(Piecewise)
+        Piecewise((DiracDelta(0), Eq(x, 0)), (0, True))
+
+        >>> DiracDelta(x - 5).rewrite(Piecewise)
+        Piecewise((DiracDelta(0), Eq(x, 5)), (0, True))
+
+        >>> DiracDelta(x**2 - 5).rewrite(Piecewise)
+           Piecewise((DiracDelta(0), Eq(x**2, 5)), (0, True))
+
+        >>> DiracDelta(x - 5, 4).rewrite(Piecewise)
+        DiracDelta(x - 5, 4)
+
+        """
+        if len(args) == 1:
+            return Piecewise((DiracDelta(0), Eq(args[0], 0)), (0, True))
+
+    def _eval_rewrite_as_SingularityFunction(self, *args, **kwargs):
+        """
+        Returns the DiracDelta expression written in the form of Singularity
+        Functions.
+
+        """
+        from sympy.solvers import solve
+        from sympy.functions.special.singularity_functions import SingularityFunction
+        if self == DiracDelta(0):
+            return SingularityFunction(0, 0, -1)
+        if self == DiracDelta(0, 1):
+            return SingularityFunction(0, 0, -2)
+        free = self.free_symbols
+        if len(free) == 1:
+            x = (free.pop())
+            if len(args) == 1:
+                return SingularityFunction(x, solve(args[0], x)[0], -1)
+            return SingularityFunction(x, solve(args[0], x)[0], -args[1] - 1)
+        else:
+            # I don't know how to handle the case for DiracDelta expressions
+            # having arguments with more than one variable.
+            raise TypeError(filldedent('''
+                rewrite(SingularityFunction) does not support
+                arguments with more that one variable.'''))
+
+
+###############################################################################
+############################## HEAVISIDE FUNCTION #############################
+###############################################################################
+
+
+class Heaviside(Function):
+    r"""
+    Heaviside step function.
+
+    Explanation
+    ===========
+
+    The Heaviside step function has the following properties:
+
+    1) $\frac{d}{d x} \theta(x) = \delta(x)$
+    2) $\theta(x) = \begin{cases} 0 & \text{for}\: x < 0 \\ \frac{1}{2} &
+       \text{for}\: x = 0 \\1 & \text{for}\: x > 0 \end{cases}$
+    3) $\frac{d}{d x} \max(x, 0) = \theta(x)$
+
+    Heaviside(x) is printed as $\theta(x)$ with the SymPy LaTeX printer.
+
+    The value at 0 is set differently in different fields. SymPy uses 1/2,
+    which is a convention from electronics and signal processing, and is
+    consistent with solving improper integrals by Fourier transform and
+    convolution.
+
+    To specify a different value of Heaviside at ``x=0``, a second argument
+    can be given. Using ``Heaviside(x, nan)`` gives an expression that will
+    evaluate to nan for x=0.
+
+    .. versionchanged:: 1.9 ``Heaviside(0)`` now returns 1/2 (before: undefined)
+
+    Examples
+    ========
+
+    >>> from sympy import Heaviside, nan
+    >>> from sympy.abc import x
+    >>> Heaviside(9)
+    1
+    >>> Heaviside(-9)
+    0
+    >>> Heaviside(0)
+    1/2
+    >>> Heaviside(0, nan)
+    nan
+    >>> (Heaviside(x) + 1).replace(Heaviside(x), Heaviside(x, 1))
+    Heaviside(x, 1) + 1
+
+    See Also
+    ========
+
+    DiracDelta
+
+    References
+    ==========
+
+    .. [1] https://mathworld.wolfram.com/HeavisideStepFunction.html
+    .. [2] https://dlmf.nist.gov/1.16#iv
+
+    """
+
+    is_real = True
+
+    def fdiff(self, argindex=1):
+        """
+        Returns the first derivative of a Heaviside Function.
+
+        Examples
+        ========
+
+        >>> from sympy import Heaviside, diff
+        >>> from sympy.abc import x
+
+        >>> Heaviside(x).fdiff()
+        DiracDelta(x)
+
+        >>> Heaviside(x**2 - 1).fdiff()
+        DiracDelta(x**2 - 1)
+
+        >>> diff(Heaviside(x)).fdiff()
+        DiracDelta(x, 1)
+
+        Parameters
+        ==========
+
+        argindex : integer
+            order of derivative
+
+        """
+        if argindex == 1:
+            return DiracDelta(self.args[0])
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def __new__(cls, arg, H0=S.Half, **options):
+        if isinstance(H0, Heaviside) and len(H0.args) == 1:
+            H0 = S.Half
+        return super(cls, cls).__new__(cls, arg, H0, **options)
+
+    @property
+    def pargs(self):
+        """Args without default S.Half"""
+        args = self.args
+        if args[1] is S.Half:
+            args = args[:1]
+        return args
+
+    @classmethod
+    def eval(cls, arg, H0=S.Half):
+        """
+        Returns a simplified form or a value of Heaviside depending on the
+        argument passed by the Heaviside object.
+
+        Explanation
+        ===========
+
+        The ``eval()`` method is automatically called when the ``Heaviside``
+        class is about to be instantiated and it returns either some simplified
+        instance or the unevaluated instance depending on the argument passed.
+        In other words, ``eval()`` method is not needed to be called explicitly,
+        it is being called and evaluated once the object is called.
+
+        Examples
+        ========
+
+        >>> from sympy import Heaviside, S
+        >>> from sympy.abc import x
+
+        >>> Heaviside(x)
+        Heaviside(x)
+
+        >>> Heaviside(19)
+        1
+
+        >>> Heaviside(0)
+        1/2
+
+        >>> Heaviside(0, 1)
+        1
+
+        >>> Heaviside(-5)
+        0
+
+        >>> Heaviside(S.NaN)
+        nan
+
+        >>> Heaviside(x - 100).subs(x, 5)
+        0
+
+        >>> Heaviside(x - 100).subs(x, 105)
+        1
+
+        Parameters
+        ==========
+
+        arg : argument passed by Heaviside object
+
+        H0 : value of Heaviside(0)
+
+        """
+        if arg.is_extended_negative:
+            return S.Zero
+        elif arg.is_extended_positive:
+            return S.One
+        elif arg.is_zero:
+            return H0
+        elif arg is S.NaN:
+            return S.NaN
+        elif fuzzy_not(im(arg).is_zero):
+            raise ValueError("Function defined only for Real Values. Complex part: %s  found in %s ." % (repr(im(arg)), repr(arg)) )
+
+    def _eval_rewrite_as_Piecewise(self, arg, H0=None, **kwargs):
+        """
+        Represents Heaviside in a Piecewise form.
+
+        Examples
+        ========
+
+        >>> from sympy import Heaviside, Piecewise, Symbol, nan
+        >>> x = Symbol('x')
+
+        >>> Heaviside(x).rewrite(Piecewise)
+        Piecewise((0, x < 0), (1/2, Eq(x, 0)), (1, True))
+
+        >>> Heaviside(x,nan).rewrite(Piecewise)
+        Piecewise((0, x < 0), (nan, Eq(x, 0)), (1, True))
+
+        >>> Heaviside(x - 5).rewrite(Piecewise)
+        Piecewise((0, x < 5), (1/2, Eq(x, 5)), (1, True))
+
+        >>> Heaviside(x**2 - 1).rewrite(Piecewise)
+        Piecewise((0, x**2 < 1), (1/2, Eq(x**2, 1)), (1, True))
+
+        """
+        if H0 == 0:
+            return Piecewise((0, arg <= 0), (1, True))
+        if H0 == 1:
+            return Piecewise((0, arg < 0), (1, True))
+        return Piecewise((0, arg < 0), (H0, Eq(arg, 0)), (1, True))
+
+    def _eval_rewrite_as_sign(self, arg, H0=S.Half, **kwargs):
+        """
+        Represents the Heaviside function in the form of sign function.
+
+        Explanation
+        ===========
+
+        The value of Heaviside(0) must be 1/2 for rewriting as sign to be
+        strictly equivalent. For easier usage, we also allow this rewriting
+        when Heaviside(0) is undefined.
+
+        Examples
+        ========
+
+        >>> from sympy import Heaviside, Symbol, sign, nan
+        >>> x = Symbol('x', real=True)
+        >>> y = Symbol('y')
+
+        >>> Heaviside(x).rewrite(sign)
+        sign(x)/2 + 1/2
+
+        >>> Heaviside(x, 0).rewrite(sign)
+        Piecewise((sign(x)/2 + 1/2, Ne(x, 0)), (0, True))
+
+        >>> Heaviside(x, nan).rewrite(sign)
+        Piecewise((sign(x)/2 + 1/2, Ne(x, 0)), (nan, True))
+
+        >>> Heaviside(x - 2).rewrite(sign)
+        sign(x - 2)/2 + 1/2
+
+        >>> Heaviside(x**2 - 2*x + 1).rewrite(sign)
+        sign(x**2 - 2*x + 1)/2 + 1/2
+
+        >>> Heaviside(y).rewrite(sign)
+        Heaviside(y)
+
+        >>> Heaviside(y**2 - 2*y + 1).rewrite(sign)
+        Heaviside(y**2 - 2*y + 1)
+
+        See Also
+        ========
+
+        sign
+
+        """
+        if arg.is_extended_real:
+            pw1 = Piecewise(
+                ((sign(arg) + 1)/2, Ne(arg, 0)),
+                (Heaviside(0, H0=H0), True))
+            pw2 = Piecewise(
+                ((sign(arg) + 1)/2, Eq(Heaviside(0, H0=H0), S.Half)),
+                (pw1, True))
+            return pw2
+
+    def _eval_rewrite_as_SingularityFunction(self, args, H0=S.Half, **kwargs):
+        """
+        Returns the Heaviside expression written in the form of Singularity
+        Functions.
+
+        """
+        from sympy.solvers import solve
+        from sympy.functions.special.singularity_functions import SingularityFunction
+        if self == Heaviside(0):
+            return SingularityFunction(0, 0, 0)
+        free = self.free_symbols
+        if len(free) == 1:
+            x = (free.pop())
+            return SingularityFunction(x, solve(args, x)[0], 0)
+            # TODO
+            # ((x - 5)**3*Heaviside(x - 5)).rewrite(SingularityFunction) should output
+            # SingularityFunction(x, 5, 0) instead of (x - 5)**3*SingularityFunction(x, 5, 0)
+        else:
+            # I don't know how to handle the case for Heaviside expressions
+            # having arguments with more than one variable.
+            raise TypeError(filldedent('''
+                rewrite(SingularityFunction) does not
+                support arguments with more that one variable.'''))

rl/Lib/site-packages/sympy/functions/special/elliptic_integrals.py

@@ -0,0 +1,445 @@
+""" Elliptic Integrals. """
+
+from sympy.core import S, pi, I, Rational
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.core.symbol import Dummy,uniquely_named_symbol
+from sympy.functions.elementary.complexes import sign
+from sympy.functions.elementary.hyperbolic import atanh
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin, tan
+from sympy.functions.special.gamma_functions import gamma
+from sympy.functions.special.hyper import hyper, meijerg
+
+class elliptic_k(Function):
+    r"""
+    The complete elliptic integral of the first kind, defined by
+
+    .. math:: K(m) = F\left(\tfrac{\pi}{2}\middle| m\right)
+
+    where $F\left(z\middle| m\right)$ is the Legendre incomplete
+    elliptic integral of the first kind.
+
+    Explanation
+    ===========
+
+    The function $K(m)$ is a single-valued function on the complex
+    plane with branch cut along the interval $(1, \infty)$.
+
+    Note that our notation defines the incomplete elliptic integral
+    in terms of the parameter $m$ instead of the elliptic modulus
+    (eccentricity) $k$.
+    In this case, the parameter $m$ is defined as $m=k^2$.
+
+    Examples
+    ========
+
+    >>> from sympy import elliptic_k, I
+    >>> from sympy.abc import m
+    >>> elliptic_k(0)
+    pi/2
+    >>> elliptic_k(1.0 + I)
+    1.50923695405127 + 0.625146415202697*I
+    >>> elliptic_k(m).series(n=3)
+    pi/2 + pi*m/8 + 9*pi*m**2/128 + O(m**3)
+
+    See Also
+    ========
+
+    elliptic_f
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
+    .. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticK
+
+    """
+
+    @classmethod
+    def eval(cls, m):
+        if m.is_zero:
+            return pi*S.Half
+        elif m is S.Half:
+            return 8*pi**Rational(3, 2)/gamma(Rational(-1, 4))**2
+        elif m is S.One:
+            return S.ComplexInfinity
+        elif m is S.NegativeOne:
+            return gamma(Rational(1, 4))**2/(4*sqrt(2*pi))
+        elif m in (S.Infinity, S.NegativeInfinity, I*S.Infinity,
+                   I*S.NegativeInfinity, S.ComplexInfinity):
+            return S.Zero
+
+    def fdiff(self, argindex=1):
+        m = self.args[0]
+        return (elliptic_e(m) - (1 - m)*elliptic_k(m))/(2*m*(1 - m))
+
+    def _eval_conjugate(self):
+        m = self.args[0]
+        if (m.is_real and (m - 1).is_positive) is False:
+            return self.func(m.conjugate())
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        from sympy.simplify import hyperexpand
+        return hyperexpand(self.rewrite(hyper)._eval_nseries(x, n=n, logx=logx))
+
+    def _eval_rewrite_as_hyper(self, m, **kwargs):
+        return pi*S.Half*hyper((S.Half, S.Half), (S.One,), m)
+
+    def _eval_rewrite_as_meijerg(self, m, **kwargs):
+        return meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -m)/2
+
+    def _eval_is_zero(self):
+        m = self.args[0]
+        if m.is_infinite:
+            return True
+
+    def _eval_rewrite_as_Integral(self, *args, **kwargs):
+        from sympy.integrals.integrals import Integral
+        t = Dummy(uniquely_named_symbol('t', args).name)
+        m = self.args[0]
+        return Integral(1/sqrt(1 - m*sin(t)**2), (t, 0, pi/2))
+
+
+class elliptic_f(Function):
+    r"""
+    The Legendre incomplete elliptic integral of the first
+    kind, defined by
+
+    .. math:: F\left(z\middle| m\right) =
+              \int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}}
+
+    Explanation
+    ===========
+
+    This function reduces to a complete elliptic integral of
+    the first kind, $K(m)$, when $z = \pi/2$.
+
+    Note that our notation defines the incomplete elliptic integral
+    in terms of the parameter $m$ instead of the elliptic modulus
+    (eccentricity) $k$.
+    In this case, the parameter $m$ is defined as $m=k^2$.
+
+    Examples
+    ========
+
+    >>> from sympy import elliptic_f, I
+    >>> from sympy.abc import z, m
+    >>> elliptic_f(z, m).series(z)
+    z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6)
+    >>> elliptic_f(3.0 + I/2, 1.0 + I)
+    2.909449841483 + 1.74720545502474*I
+
+    See Also
+    ========
+
+    elliptic_k
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
+    .. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticF
+
+    """
+
+    @classmethod
+    def eval(cls, z, m):
+        if z.is_zero:
+            return S.Zero
+        if m.is_zero:
+            return z
+        k = 2*z/pi
+        if k.is_integer:
+            return k*elliptic_k(m)
+        elif m in (S.Infinity, S.NegativeInfinity):
+            return S.Zero
+        elif z.could_extract_minus_sign():
+            return -elliptic_f(-z, m)
+
+    def fdiff(self, argindex=1):
+        z, m = self.args
+        fm = sqrt(1 - m*sin(z)**2)
+        if argindex == 1:
+            return 1/fm
+        elif argindex == 2:
+            return (elliptic_e(z, m)/(2*m*(1 - m)) - elliptic_f(z, m)/(2*m) -
+                    sin(2*z)/(4*(1 - m)*fm))
+        raise ArgumentIndexError(self, argindex)
+
+    def _eval_conjugate(self):
+        z, m = self.args
+        if (m.is_real and (m - 1).is_positive) is False:
+            return self.func(z.conjugate(), m.conjugate())
+
+    def _eval_rewrite_as_Integral(self, *args, **kwargs):
+        from sympy.integrals.integrals import Integral
+        t = Dummy(uniquely_named_symbol('t', args).name)
+        z, m = self.args[0], self.args[1]
+        return Integral(1/(sqrt(1 - m*sin(t)**2)), (t, 0, z))
+
+    def _eval_is_zero(self):
+        z, m = self.args
+        if z.is_zero:
+            return True
+        if m.is_extended_real and m.is_infinite:
+            return True
+
+
+class elliptic_e(Function):
+    r"""
+    Called with two arguments $z$ and $m$, evaluates the
+    incomplete elliptic integral of the second kind, defined by
+
+    .. math:: E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt
+
+    Called with a single argument $m$, evaluates the Legendre complete
+    elliptic integral of the second kind
+
+    .. math:: E(m) = E\left(\tfrac{\pi}{2}\middle| m\right)
+
+    Explanation
+    ===========
+
+    The function $E(m)$ is a single-valued function on the complex
+    plane with branch cut along the interval $(1, \infty)$.
+
+    Note that our notation defines the incomplete elliptic integral
+    in terms of the parameter $m$ instead of the elliptic modulus
+    (eccentricity) $k$.
+    In this case, the parameter $m$ is defined as $m=k^2$.
+
+    Examples
+    ========
+
+    >>> from sympy import elliptic_e, I
+    >>> from sympy.abc import z, m
+    >>> elliptic_e(z, m).series(z)
+    z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)
+    >>> elliptic_e(m).series(n=4)
+    pi/2 - pi*m/8 - 3*pi*m**2/128 - 5*pi*m**3/512 + O(m**4)
+    >>> elliptic_e(1 + I, 2 - I/2).n()
+    1.55203744279187 + 0.290764986058437*I
+    >>> elliptic_e(0)
+    pi/2
+    >>> elliptic_e(2.0 - I)
+    0.991052601328069 + 0.81879421395609*I
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
+    .. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticE2
+    .. [3] https://functions.wolfram.com/EllipticIntegrals/EllipticE
+
+    """
+
+    @classmethod
+    def eval(cls, m, z=None):
+        if z is not None:
+            z, m = m, z
+            k = 2*z/pi
+            if m.is_zero:
+                return z
+            if z.is_zero:
+                return S.Zero
+            elif k.is_integer:
+                return k*elliptic_e(m)
+            elif m in (S.Infinity, S.NegativeInfinity):
+                return S.ComplexInfinity
+            elif z.could_extract_minus_sign():
+                return -elliptic_e(-z, m)
+        else:
+            if m.is_zero:
+                return pi/2
+            elif m is S.One:
+                return S.One
+            elif m is S.Infinity:
+                return I*S.Infinity
+            elif m is S.NegativeInfinity:
+                return S.Infinity
+            elif m is S.ComplexInfinity:
+                return S.ComplexInfinity
+
+    def fdiff(self, argindex=1):
+        if len(self.args) == 2:
+            z, m = self.args
+            if argindex == 1:
+                return sqrt(1 - m*sin(z)**2)
+            elif argindex == 2:
+                return (elliptic_e(z, m) - elliptic_f(z, m))/(2*m)
+        else:
+            m = self.args[0]
+            if argindex == 1:
+                return (elliptic_e(m) - elliptic_k(m))/(2*m)
+        raise ArgumentIndexError(self, argindex)
+
+    def _eval_conjugate(self):
+        if len(self.args) == 2:
+            z, m = self.args
+            if (m.is_real and (m - 1).is_positive) is False:
+                return self.func(z.conjugate(), m.conjugate())
+        else:
+            m = self.args[0]
+            if (m.is_real and (m - 1).is_positive) is False:
+                return self.func(m.conjugate())
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        from sympy.simplify import hyperexpand
+        if len(self.args) == 1:
+            return hyperexpand(self.rewrite(hyper)._eval_nseries(x, n=n, logx=logx))
+        return super()._eval_nseries(x, n=n, logx=logx)
+
+    def _eval_rewrite_as_hyper(self, *args, **kwargs):
+        if len(args) == 1:
+            m = args[0]
+            return (pi/2)*hyper((Rational(-1, 2), S.Half), (S.One,), m)
+
+    def _eval_rewrite_as_meijerg(self, *args, **kwargs):
+        if len(args) == 1:
+            m = args[0]
+            return -meijerg(((S.Half, Rational(3, 2)), []), \
+                            ((S.Zero,), (S.Zero,)), -m)/4
+
+    def _eval_rewrite_as_Integral(self, *args, **kwargs):
+        from sympy.integrals.integrals import Integral
+        z, m = (pi/2, self.args[0]) if len(self.args) == 1 else self.args
+        t = Dummy(uniquely_named_symbol('t', args).name)
+        return Integral(sqrt(1 - m*sin(t)**2), (t, 0, z))
+
+
+class elliptic_pi(Function):
+    r"""
+    Called with three arguments $n$, $z$ and $m$, evaluates the
+    Legendre incomplete elliptic integral of the third kind, defined by
+
+    .. math:: \Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt}
+              {\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}}
+
+    Called with two arguments $n$ and $m$, evaluates the complete
+    elliptic integral of the third kind:
+
+    .. math:: \Pi\left(n\middle| m\right) =
+              \Pi\left(n; \tfrac{\pi}{2}\middle| m\right)
+
+    Explanation
+    ===========
+
+    Note that our notation defines the incomplete elliptic integral
+    in terms of the parameter $m$ instead of the elliptic modulus
+    (eccentricity) $k$.
+    In this case, the parameter $m$ is defined as $m=k^2$.
+
+    Examples
+    ========
+
+    >>> from sympy import elliptic_pi, I
+    >>> from sympy.abc import z, n, m
+    >>> elliptic_pi(n, z, m).series(z, n=4)
+    z + z**3*(m/6 + n/3) + O(z**4)
+    >>> elliptic_pi(0.5 + I, 1.0 - I, 1.2)
+    2.50232379629182 - 0.760939574180767*I
+    >>> elliptic_pi(0, 0)
+    pi/2
+    >>> elliptic_pi(1.0 - I/3, 2.0 + I)
+    3.29136443417283 + 0.32555634906645*I
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
+    .. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticPi3
+    .. [3] https://functions.wolfram.com/EllipticIntegrals/EllipticPi
+
+    """
+
+    @classmethod
+    def eval(cls, n, m, z=None):
+        if z is not None:
+            n, z, m = n, m, z
+            if n.is_zero:
+                return elliptic_f(z, m)
+            elif n is S.One:
+                return (elliptic_f(z, m) +
+                        (sqrt(1 - m*sin(z)**2)*tan(z) -
+                         elliptic_e(z, m))/(1 - m))
+            k = 2*z/pi
+            if k.is_integer:
+                return k*elliptic_pi(n, m)
+            elif m.is_zero:
+                return atanh(sqrt(n - 1)*tan(z))/sqrt(n - 1)
+            elif n == m:
+                return (elliptic_f(z, n) - elliptic_pi(1, z, n) +
+                        tan(z)/sqrt(1 - n*sin(z)**2))
+            elif n in (S.Infinity, S.NegativeInfinity):
+                return S.Zero
+            elif m in (S.Infinity, S.NegativeInfinity):
+                return S.Zero
+            elif z.could_extract_minus_sign():
+                return -elliptic_pi(n, -z, m)
+            if n.is_zero:
+                return elliptic_f(z, m)
+            if m.is_extended_real and m.is_infinite or \
+                    n.is_extended_real and n.is_infinite:
+                return S.Zero
+        else:
+            if n.is_zero:
+                return elliptic_k(m)
+            elif n is S.One:
+                return S.ComplexInfinity
+            elif m.is_zero:
+                return pi/(2*sqrt(1 - n))
+            elif m == S.One:
+                return S.NegativeInfinity/sign(n - 1)
+            elif n == m:
+                return elliptic_e(n)/(1 - n)
+            elif n in (S.Infinity, S.NegativeInfinity):
+                return S.Zero
+            elif m in (S.Infinity, S.NegativeInfinity):
+                return S.Zero
+            if n.is_zero:
+                return elliptic_k(m)
+            if m.is_extended_real and m.is_infinite or \
+                    n.is_extended_real and n.is_infinite:
+                return S.Zero
+
+    def _eval_conjugate(self):
+        if len(self.args) == 3:
+            n, z, m = self.args
+            if (n.is_real and (n - 1).is_positive) is False and \
+               (m.is_real and (m - 1).is_positive) is False:
+                return self.func(n.conjugate(), z.conjugate(), m.conjugate())
+        else:
+            n, m = self.args
+            return self.func(n.conjugate(), m.conjugate())
+
+    def fdiff(self, argindex=1):
+        if len(self.args) == 3:
+            n, z, m = self.args
+            fm, fn = sqrt(1 - m*sin(z)**2), 1 - n*sin(z)**2
+            if argindex == 1:
+                return (elliptic_e(z, m) + (m - n)*elliptic_f(z, m)/n +
+                        (n**2 - m)*elliptic_pi(n, z, m)/n -
+                        n*fm*sin(2*z)/(2*fn))/(2*(m - n)*(n - 1))
+            elif argindex == 2:
+                return 1/(fm*fn)
+            elif argindex == 3:
+                return (elliptic_e(z, m)/(m - 1) +
+                        elliptic_pi(n, z, m) -
+                        m*sin(2*z)/(2*(m - 1)*fm))/(2*(n - m))
+        else:
+            n, m = self.args
+            if argindex == 1:
+                return (elliptic_e(m) + (m - n)*elliptic_k(m)/n +
+                        (n**2 - m)*elliptic_pi(n, m)/n)/(2*(m - n)*(n - 1))
+            elif argindex == 2:
+                return (elliptic_e(m)/(m - 1) + elliptic_pi(n, m))/(2*(n - m))
+        raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Integral(self, *args, **kwargs):
+        from sympy.integrals.integrals import Integral
+        if len(self.args) == 2:
+            n, m, z = self.args[0], self.args[1], pi/2
+        else:
+            n, z, m = self.args
+        t = Dummy(uniquely_named_symbol('t', args).name)
+        return Integral(1/((1 - n*sin(t)**2)*sqrt(1 - m*sin(t)**2)), (t, 0, z))

rl/Lib/site-packages/sympy/functions/special/mathieu_functions.py

@@ -0,0 +1,269 @@
+""" This module contains the Mathieu functions.
+"""
+
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin, cos
+
+
+class MathieuBase(Function):
+    """
+    Abstract base class for Mathieu functions.
+
+    This class is meant to reduce code duplication.
+
+    """
+
+    unbranched = True
+
+    def _eval_conjugate(self):
+        a, q, z = self.args
+        return self.func(a.conjugate(), q.conjugate(), z.conjugate())
+
+
+class mathieus(MathieuBase):
+    r"""
+    The Mathieu Sine function $S(a,q,z)$.
+
+    Explanation
+    ===========
+
+    This function is one solution of the Mathieu differential equation:
+
+    .. math ::
+        y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0
+
+    The other solution is the Mathieu Cosine function.
+
+    Examples
+    ========
+
+    >>> from sympy import diff, mathieus
+    >>> from sympy.abc import a, q, z
+
+    >>> mathieus(a, q, z)
+    mathieus(a, q, z)
+
+    >>> mathieus(a, 0, z)
+    sin(sqrt(a)*z)
+
+    >>> diff(mathieus(a, q, z), z)
+    mathieusprime(a, q, z)
+
+    See Also
+    ========
+
+    mathieuc: Mathieu cosine function.
+    mathieusprime: Derivative of Mathieu sine function.
+    mathieucprime: Derivative of Mathieu cosine function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Mathieu_function
+    .. [2] https://dlmf.nist.gov/28
+    .. [3] https://mathworld.wolfram.com/MathieuFunction.html
+    .. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuS/
+
+    """
+
+    def fdiff(self, argindex=1):
+        if argindex == 3:
+            a, q, z = self.args
+            return mathieusprime(a, q, z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, a, q, z):
+        if q.is_Number and q.is_zero:
+            return sin(sqrt(a)*z)
+        # Try to pull out factors of -1
+        if z.could_extract_minus_sign():
+            return -cls(a, q, -z)
+
+
+class mathieuc(MathieuBase):
+    r"""
+    The Mathieu Cosine function $C(a,q,z)$.
+
+    Explanation
+    ===========
+
+    This function is one solution of the Mathieu differential equation:
+
+    .. math ::
+        y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0
+
+    The other solution is the Mathieu Sine function.
+
+    Examples
+    ========
+
+    >>> from sympy import diff, mathieuc
+    >>> from sympy.abc import a, q, z
+
+    >>> mathieuc(a, q, z)
+    mathieuc(a, q, z)
+
+    >>> mathieuc(a, 0, z)
+    cos(sqrt(a)*z)
+
+    >>> diff(mathieuc(a, q, z), z)
+    mathieucprime(a, q, z)
+
+    See Also
+    ========
+
+    mathieus: Mathieu sine function
+    mathieusprime: Derivative of Mathieu sine function
+    mathieucprime: Derivative of Mathieu cosine function
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Mathieu_function
+    .. [2] https://dlmf.nist.gov/28
+    .. [3] https://mathworld.wolfram.com/MathieuFunction.html
+    .. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuC/
+
+    """
+
+    def fdiff(self, argindex=1):
+        if argindex == 3:
+            a, q, z = self.args
+            return mathieucprime(a, q, z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, a, q, z):
+        if q.is_Number and q.is_zero:
+            return cos(sqrt(a)*z)
+        # Try to pull out factors of -1
+        if z.could_extract_minus_sign():
+            return cls(a, q, -z)
+
+
+class mathieusprime(MathieuBase):
+    r"""
+    The derivative $S^{\prime}(a,q,z)$ of the Mathieu Sine function.
+
+    Explanation
+    ===========
+
+    This function is one solution of the Mathieu differential equation:
+
+    .. math ::
+        y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0
+
+    The other solution is the Mathieu Cosine function.
+
+    Examples
+    ========
+
+    >>> from sympy import diff, mathieusprime
+    >>> from sympy.abc import a, q, z
+
+    >>> mathieusprime(a, q, z)
+    mathieusprime(a, q, z)
+
+    >>> mathieusprime(a, 0, z)
+    sqrt(a)*cos(sqrt(a)*z)
+
+    >>> diff(mathieusprime(a, q, z), z)
+    (-a + 2*q*cos(2*z))*mathieus(a, q, z)
+
+    See Also
+    ========
+
+    mathieus: Mathieu sine function
+    mathieuc: Mathieu cosine function
+    mathieucprime: Derivative of Mathieu cosine function
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Mathieu_function
+    .. [2] https://dlmf.nist.gov/28
+    .. [3] https://mathworld.wolfram.com/MathieuFunction.html
+    .. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuSPrime/
+
+    """
+
+    def fdiff(self, argindex=1):
+        if argindex == 3:
+            a, q, z = self.args
+            return (2*q*cos(2*z) - a)*mathieus(a, q, z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, a, q, z):
+        if q.is_Number and q.is_zero:
+            return sqrt(a)*cos(sqrt(a)*z)
+        # Try to pull out factors of -1
+        if z.could_extract_minus_sign():
+            return cls(a, q, -z)
+
+
+class mathieucprime(MathieuBase):
+    r"""
+    The derivative $C^{\prime}(a,q,z)$ of the Mathieu Cosine function.
+
+    Explanation
+    ===========
+
+    This function is one solution of the Mathieu differential equation:
+
+    .. math ::
+        y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0
+
+    The other solution is the Mathieu Sine function.
+
+    Examples
+    ========
+
+    >>> from sympy import diff, mathieucprime
+    >>> from sympy.abc import a, q, z
+
+    >>> mathieucprime(a, q, z)
+    mathieucprime(a, q, z)
+
+    >>> mathieucprime(a, 0, z)
+    -sqrt(a)*sin(sqrt(a)*z)
+
+    >>> diff(mathieucprime(a, q, z), z)
+    (-a + 2*q*cos(2*z))*mathieuc(a, q, z)
+
+    See Also
+    ========
+
+    mathieus: Mathieu sine function
+    mathieuc: Mathieu cosine function
+    mathieusprime: Derivative of Mathieu sine function
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Mathieu_function
+    .. [2] https://dlmf.nist.gov/28
+    .. [3] https://mathworld.wolfram.com/MathieuFunction.html
+    .. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuCPrime/
+
+    """
+
+    def fdiff(self, argindex=1):
+        if argindex == 3:
+            a, q, z = self.args
+            return (2*q*cos(2*z) - a)*mathieuc(a, q, z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, a, q, z):
+        if q.is_Number and q.is_zero:
+            return -sqrt(a)*sin(sqrt(a)*z)
+        # Try to pull out factors of -1
+        if z.could_extract_minus_sign():
+            return -cls(a, q, -z)

rl/Lib/site-packages/sympy/functions/special/singularity_functions.py

@@ -0,0 +1,235 @@
+from sympy.core import S, oo, diff
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.core.logic import fuzzy_not
+from sympy.core.relational import Eq
+from sympy.functions.elementary.complexes import im
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.special.delta_functions import Heaviside
+
+###############################################################################
+############################# SINGULARITY FUNCTION ############################
+###############################################################################
+
+
+class SingularityFunction(Function):
+    r"""
+    Singularity functions are a class of discontinuous functions.
+
+    Explanation
+    ===========
+
+    Singularity functions take a variable, an offset, and an exponent as
+    arguments. These functions are represented using Macaulay brackets as:
+
+    SingularityFunction(x, a, n) := <x - a>^n
+
+    The singularity function will automatically evaluate to
+    ``Derivative(DiracDelta(x - a), x, -n - 1)`` if ``n < 0``
+    and ``(x - a)**n*Heaviside(x - a, 1)`` if ``n >= 0``.
+
+    Examples
+    ========
+
+    >>> from sympy import SingularityFunction, diff, Piecewise, DiracDelta, Heaviside, Symbol
+    >>> from sympy.abc import x, a, n
+    >>> SingularityFunction(x, a, n)
+    SingularityFunction(x, a, n)
+    >>> y = Symbol('y', positive=True)
+    >>> n = Symbol('n', nonnegative=True)
+    >>> SingularityFunction(y, -10, n)
+    (y + 10)**n
+    >>> y = Symbol('y', negative=True)
+    >>> SingularityFunction(y, 10, n)
+    0
+    >>> SingularityFunction(x, 4, -1).subs(x, 4)
+    oo
+    >>> SingularityFunction(x, 10, -2).subs(x, 10)
+    oo
+    >>> SingularityFunction(4, 1, 5)
+    243
+    >>> diff(SingularityFunction(x, 1, 5) + SingularityFunction(x, 1, 4), x)
+    4*SingularityFunction(x, 1, 3) + 5*SingularityFunction(x, 1, 4)
+    >>> diff(SingularityFunction(x, 4, 0), x, 2)
+    SingularityFunction(x, 4, -2)
+    >>> SingularityFunction(x, 4, 5).rewrite(Piecewise)
+    Piecewise(((x - 4)**5, x >= 4), (0, True))
+    >>> expr = SingularityFunction(x, a, n)
+    >>> y = Symbol('y', positive=True)
+    >>> n = Symbol('n', nonnegative=True)
+    >>> expr.subs({x: y, a: -10, n: n})
+    (y + 10)**n
+
+    The methods ``rewrite(DiracDelta)``, ``rewrite(Heaviside)``, and
+    ``rewrite('HeavisideDiracDelta')`` returns the same output. One can use any
+    of these methods according to their choice.
+
+    >>> expr = SingularityFunction(x, 4, 5) + SingularityFunction(x, -3, -1) - SingularityFunction(x, 0, -2)
+    >>> expr.rewrite(Heaviside)
+    (x - 4)**5*Heaviside(x - 4, 1) + DiracDelta(x + 3) - DiracDelta(x, 1)
+    >>> expr.rewrite(DiracDelta)
+    (x - 4)**5*Heaviside(x - 4, 1) + DiracDelta(x + 3) - DiracDelta(x, 1)
+    >>> expr.rewrite('HeavisideDiracDelta')
+    (x - 4)**5*Heaviside(x - 4, 1) + DiracDelta(x + 3) - DiracDelta(x, 1)
+
+    See Also
+    ========
+
+    DiracDelta, Heaviside
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Singularity_function
+
+    """
+
+    is_real = True
+
+    def fdiff(self, argindex=1):
+        """
+        Returns the first derivative of a DiracDelta Function.
+
+        Explanation
+        ===========
+
+        The difference between ``diff()`` and ``fdiff()`` is: ``diff()`` is the
+        user-level function and ``fdiff()`` is an object method. ``fdiff()`` is
+        a convenience method available in the ``Function`` class. It returns
+        the derivative of the function without considering the chain rule.
+        ``diff(function, x)`` calls ``Function._eval_derivative`` which in turn
+        calls ``fdiff()`` internally to compute the derivative of the function.
+
+        """
+
+        if argindex == 1:
+            x, a, n = self.args
+            if n in (S.Zero, S.NegativeOne, S(-2), S(-3)):
+                return self.func(x, a, n-1)
+            elif n.is_positive:
+                return n*self.func(x, a, n-1)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, variable, offset, exponent):
+        """
+        Returns a simplified form or a value of Singularity Function depending
+        on the argument passed by the object.
+
+        Explanation
+        ===========
+
+        The ``eval()`` method is automatically called when the
+        ``SingularityFunction`` class is about to be instantiated and it
+        returns either some simplified instance or the unevaluated instance
+        depending on the argument passed. In other words, ``eval()`` method is
+        not needed to be called explicitly, it is being called and evaluated
+        once the object is called.
+
+        Examples
+        ========
+
+        >>> from sympy import SingularityFunction, Symbol, nan
+        >>> from sympy.abc import x, a, n
+        >>> SingularityFunction(x, a, n)
+        SingularityFunction(x, a, n)
+        >>> SingularityFunction(5, 3, 2)
+        4
+        >>> SingularityFunction(x, a, nan)
+        nan
+        >>> SingularityFunction(x, 3, 0).subs(x, 3)
+        1
+        >>> SingularityFunction(4, 1, 5)
+        243
+        >>> x = Symbol('x', positive = True)
+        >>> a = Symbol('a', negative = True)
+        >>> n = Symbol('n', nonnegative = True)
+        >>> SingularityFunction(x, a, n)
+        (-a + x)**n
+        >>> x = Symbol('x', negative = True)
+        >>> a = Symbol('a', positive = True)
+        >>> SingularityFunction(x, a, n)
+        0
+
+        """
+
+        x = variable
+        a = offset
+        n = exponent
+        shift = (x - a)
+
+        if fuzzy_not(im(shift).is_zero):
+            raise ValueError("Singularity Functions are defined only for Real Numbers.")
+        if fuzzy_not(im(n).is_zero):
+            raise ValueError("Singularity Functions are not defined for imaginary exponents.")
+        if shift is S.NaN or n is S.NaN:
+            return S.NaN
+        if (n + 4).is_negative:
+            raise ValueError("Singularity Functions are not defined for exponents less than -4.")
+        if shift.is_extended_negative:
+            return S.Zero
+        if n.is_nonnegative:
+            if shift.is_zero:  # use literal 0 in case of Symbol('z', zero=True)
+                return S.Zero**n
+            if shift.is_extended_nonnegative:
+                return shift**n
+        if n in (S.NegativeOne, -2, -3, -4):
+            if shift.is_negative or shift.is_extended_positive:
+                return S.Zero
+            if shift.is_zero:
+                return oo
+
+    def _eval_rewrite_as_Piecewise(self, *args, **kwargs):
+        '''
+        Converts a Singularity Function expression into its Piecewise form.
+
+        '''
+        x, a, n = self.args
+
+        if n in (S.NegativeOne, S(-2), S(-3), S(-4)):
+            return Piecewise((oo, Eq(x - a, 0)), (0, True))
+        elif n.is_nonnegative:
+            return Piecewise(((x - a)**n, x - a >= 0), (0, True))
+
+    def _eval_rewrite_as_Heaviside(self, *args, **kwargs):
+        '''
+        Rewrites a Singularity Function expression using Heavisides and DiracDeltas.
+
+        '''
+        x, a, n = self.args
+
+        if n == -4:
+            return diff(Heaviside(x - a), x.free_symbols.pop(), 4)
+        if n == -3:
+            return diff(Heaviside(x - a), x.free_symbols.pop(), 3)
+        if n == -2:
+            return diff(Heaviside(x - a), x.free_symbols.pop(), 2)
+        if n == -1:
+            return diff(Heaviside(x - a), x.free_symbols.pop(), 1)
+        if n.is_nonnegative:
+            return (x - a)**n*Heaviside(x - a, 1)
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        z, a, n = self.args
+        shift = (z - a).subs(x, 0)
+        if n < 0:
+            return S.Zero
+        elif n.is_zero and shift.is_zero:
+            return S.Zero if cdir == -1 else S.One
+        elif shift.is_positive:
+            return shift**n
+        return S.Zero
+
+    def _eval_nseries(self, x, n, logx=None, cdir=0):
+        z, a, n = self.args
+        shift = (z - a).subs(x, 0)
+        if n < 0:
+            return S.Zero
+        elif n.is_zero and shift.is_zero:
+            return S.Zero if cdir == -1 else S.One
+        elif shift.is_positive:
+            return ((z - a)**n)._eval_nseries(x, n, logx=logx, cdir=cdir)
+        return S.Zero
+
+    _eval_rewrite_as_DiracDelta = _eval_rewrite_as_Heaviside
+    _eval_rewrite_as_HeavisideDiracDelta = _eval_rewrite_as_Heaviside

rl/Lib/site-packages/sympy/functions/special/spherical_harmonics.py

@@ -0,0 +1,334 @@
+from sympy.core.expr import Expr
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.core.numbers import I, pi
+from sympy.core.singleton import S
+from sympy.core.symbol import Dummy
+from sympy.functions import assoc_legendre
+from sympy.functions.combinatorial.factorials import factorial
+from sympy.functions.elementary.complexes import Abs, conjugate
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin, cos, cot
+
+_x = Dummy("x")
+
+class Ynm(Function):
+    r"""
+    Spherical harmonics defined as
+
+    .. math::
+        Y_n^m(\theta, \varphi) := \sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}}
+                                  \exp(i m \varphi)
+                                  \mathrm{P}_n^m\left(\cos(\theta)\right)
+
+    Explanation
+    ===========
+
+    ``Ynm()`` gives the spherical harmonic function of order $n$ and $m$
+    in $\theta$ and $\varphi$, $Y_n^m(\theta, \varphi)$. The four
+    parameters are as follows: $n \geq 0$ an integer and $m$ an integer
+    such that $-n \leq m \leq n$ holds. The two angles are real-valued
+    with $\theta \in [0, \pi]$ and $\varphi \in [0, 2\pi]$.
+
+    Examples
+    ========
+
+    >>> from sympy import Ynm, Symbol, simplify
+    >>> from sympy.abc import n,m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+
+    >>> Ynm(n, m, theta, phi)
+    Ynm(n, m, theta, phi)
+
+    Several symmetries are known, for the order:
+
+    >>> Ynm(n, -m, theta, phi)
+    (-1)**m*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
+
+    As well as for the angles:
+
+    >>> Ynm(n, m, -theta, phi)
+    Ynm(n, m, theta, phi)
+
+    >>> Ynm(n, m, theta, -phi)
+    exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
+
+    For specific integers $n$ and $m$ we can evaluate the harmonics
+    to more useful expressions:
+
+    >>> simplify(Ynm(0, 0, theta, phi).expand(func=True))
+    1/(2*sqrt(pi))
+
+    >>> simplify(Ynm(1, -1, theta, phi).expand(func=True))
+    sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))
+
+    >>> simplify(Ynm(1, 0, theta, phi).expand(func=True))
+    sqrt(3)*cos(theta)/(2*sqrt(pi))
+
+    >>> simplify(Ynm(1, 1, theta, phi).expand(func=True))
+    -sqrt(6)*exp(I*phi)*sin(theta)/(4*sqrt(pi))
+
+    >>> simplify(Ynm(2, -2, theta, phi).expand(func=True))
+    sqrt(30)*exp(-2*I*phi)*sin(theta)**2/(8*sqrt(pi))
+
+    >>> simplify(Ynm(2, -1, theta, phi).expand(func=True))
+    sqrt(30)*exp(-I*phi)*sin(2*theta)/(8*sqrt(pi))
+
+    >>> simplify(Ynm(2, 0, theta, phi).expand(func=True))
+    sqrt(5)*(3*cos(theta)**2 - 1)/(4*sqrt(pi))
+
+    >>> simplify(Ynm(2, 1, theta, phi).expand(func=True))
+    -sqrt(30)*exp(I*phi)*sin(2*theta)/(8*sqrt(pi))
+
+    >>> simplify(Ynm(2, 2, theta, phi).expand(func=True))
+    sqrt(30)*exp(2*I*phi)*sin(theta)**2/(8*sqrt(pi))
+
+    We can differentiate the functions with respect
+    to both angles:
+
+    >>> from sympy import Ynm, Symbol, diff
+    >>> from sympy.abc import n,m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+
+    >>> diff(Ynm(n, m, theta, phi), theta)
+    m*cot(theta)*Ynm(n, m, theta, phi) + sqrt((-m + n)*(m + n + 1))*exp(-I*phi)*Ynm(n, m + 1, theta, phi)
+
+    >>> diff(Ynm(n, m, theta, phi), phi)
+    I*m*Ynm(n, m, theta, phi)
+
+    Further we can compute the complex conjugation:
+
+    >>> from sympy import Ynm, Symbol, conjugate
+    >>> from sympy.abc import n,m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+
+    >>> conjugate(Ynm(n, m, theta, phi))
+    (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
+
+    To get back the well known expressions in spherical
+    coordinates, we use full expansion:
+
+    >>> from sympy import Ynm, Symbol, expand_func
+    >>> from sympy.abc import n,m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+
+    >>> expand_func(Ynm(n, m, theta, phi))
+    sqrt((2*n + 1)*factorial(-m + n)/factorial(m + n))*exp(I*m*phi)*assoc_legendre(n, m, cos(theta))/(2*sqrt(pi))
+
+    See Also
+    ========
+
+    Ynm_c, Znm
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
+    .. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
+    .. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/
+    .. [4] https://dlmf.nist.gov/14.30
+
+    """
+
+    @classmethod
+    def eval(cls, n, m, theta, phi):
+        # Handle negative index m and arguments theta, phi
+        if m.could_extract_minus_sign():
+            m = -m
+            return S.NegativeOne**m * exp(-2*I*m*phi) * Ynm(n, m, theta, phi)
+        if theta.could_extract_minus_sign():
+            theta = -theta
+            return Ynm(n, m, theta, phi)
+        if phi.could_extract_minus_sign():
+            phi = -phi
+            return exp(-2*I*m*phi) * Ynm(n, m, theta, phi)
+
+        # TODO Add more simplififcation here
+
+    def _eval_expand_func(self, **hints):
+        n, m, theta, phi = self.args
+        rv = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
+                exp(I*m*phi) * assoc_legendre(n, m, cos(theta)))
+        # We can do this because of the range of theta
+        return rv.subs(sqrt(-cos(theta)**2 + 1), sin(theta))
+
+    def fdiff(self, argindex=4):
+        if argindex == 1:
+            # Diff wrt n
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 2:
+            # Diff wrt m
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 3:
+            # Diff wrt theta
+            n, m, theta, phi = self.args
+            return (m * cot(theta) * Ynm(n, m, theta, phi) +
+                    sqrt((n - m)*(n + m + 1)) * exp(-I*phi) * Ynm(n, m + 1, theta, phi))
+        elif argindex == 4:
+            # Diff wrt phi
+            n, m, theta, phi = self.args
+            return I * m * Ynm(n, m, theta, phi)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_polynomial(self, n, m, theta, phi, **kwargs):
+        # TODO: Make sure n \in N
+        # TODO: Assert |m| <= n ortherwise we should return 0
+        return self.expand(func=True)
+
+    def _eval_rewrite_as_sin(self, n, m, theta, phi, **kwargs):
+        return self.rewrite(cos)
+
+    def _eval_rewrite_as_cos(self, n, m, theta, phi, **kwargs):
+        # This method can be expensive due to extensive use of simplification!
+        from sympy.simplify import simplify, trigsimp
+        # TODO: Make sure n \in N
+        # TODO: Assert |m| <= n ortherwise we should return 0
+        term = simplify(self.expand(func=True))
+        # We can do this because of the range of theta
+        term = term.xreplace({Abs(sin(theta)):sin(theta)})
+        return simplify(trigsimp(term))
+
+    def _eval_conjugate(self):
+        # TODO: Make sure theta \in R and phi \in R
+        n, m, theta, phi = self.args
+        return S.NegativeOne**m * self.func(n, -m, theta, phi)
+
+    def as_real_imag(self, deep=True, **hints):
+        # TODO: Handle deep and hints
+        n, m, theta, phi = self.args
+        re = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
+              cos(m*phi) * assoc_legendre(n, m, cos(theta)))
+        im = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
+              sin(m*phi) * assoc_legendre(n, m, cos(theta)))
+        return (re, im)
+
+    def _eval_evalf(self, prec):
+        # Note: works without this function by just calling
+        #       mpmath for Legendre polynomials. But using
+        #       the dedicated function directly is cleaner.
+        from mpmath import mp, workprec
+        n = self.args[0]._to_mpmath(prec)
+        m = self.args[1]._to_mpmath(prec)
+        theta = self.args[2]._to_mpmath(prec)
+        phi = self.args[3]._to_mpmath(prec)
+        with workprec(prec):
+            res = mp.spherharm(n, m, theta, phi)
+        return Expr._from_mpmath(res, prec)
+
+
+def Ynm_c(n, m, theta, phi):
+    r"""
+    Conjugate spherical harmonics defined as
+
+    .. math::
+        \overline{Y_n^m(\theta, \varphi)} := (-1)^m Y_n^{-m}(\theta, \varphi).
+
+    Examples
+    ========
+
+    >>> from sympy import Ynm_c, Symbol, simplify
+    >>> from sympy.abc import n,m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+    >>> Ynm_c(n, m, theta, phi)
+    (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
+    >>> Ynm_c(n, m, -theta, phi)
+    (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
+
+    For specific integers $n$ and $m$ we can evaluate the harmonics
+    to more useful expressions:
+
+    >>> simplify(Ynm_c(0, 0, theta, phi).expand(func=True))
+    1/(2*sqrt(pi))
+    >>> simplify(Ynm_c(1, -1, theta, phi).expand(func=True))
+    sqrt(6)*exp(I*(-phi + 2*conjugate(phi)))*sin(theta)/(4*sqrt(pi))
+
+    See Also
+    ========
+
+    Ynm, Znm
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
+    .. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
+    .. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/
+
+    """
+    return conjugate(Ynm(n, m, theta, phi))
+
+
+class Znm(Function):
+    r"""
+    Real spherical harmonics defined as
+
+    .. math::
+
+        Z_n^m(\theta, \varphi) :=
+        \begin{cases}
+          \frac{Y_n^m(\theta, \varphi) + \overline{Y_n^m(\theta, \varphi)}}{\sqrt{2}} &\quad m > 0 \\
+          Y_n^m(\theta, \varphi) &\quad m = 0 \\
+          \frac{Y_n^m(\theta, \varphi) - \overline{Y_n^m(\theta, \varphi)}}{i \sqrt{2}} &\quad m < 0 \\
+        \end{cases}
+
+    which gives in simplified form
+
+    .. math::
+
+        Z_n^m(\theta, \varphi) =
+        \begin{cases}
+          \frac{Y_n^m(\theta, \varphi) + (-1)^m Y_n^{-m}(\theta, \varphi)}{\sqrt{2}} &\quad m > 0 \\
+          Y_n^m(\theta, \varphi) &\quad m = 0 \\
+          \frac{Y_n^m(\theta, \varphi) - (-1)^m Y_n^{-m}(\theta, \varphi)}{i \sqrt{2}} &\quad m < 0 \\
+        \end{cases}
+
+    Examples
+    ========
+
+    >>> from sympy import Znm, Symbol, simplify
+    >>> from sympy.abc import n, m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+    >>> Znm(n, m, theta, phi)
+    Znm(n, m, theta, phi)
+
+    For specific integers n and m we can evaluate the harmonics
+    to more useful expressions:
+
+    >>> simplify(Znm(0, 0, theta, phi).expand(func=True))
+    1/(2*sqrt(pi))
+    >>> simplify(Znm(1, 1, theta, phi).expand(func=True))
+    -sqrt(3)*sin(theta)*cos(phi)/(2*sqrt(pi))
+    >>> simplify(Znm(2, 1, theta, phi).expand(func=True))
+    -sqrt(15)*sin(2*theta)*cos(phi)/(4*sqrt(pi))
+
+    See Also
+    ========
+
+    Ynm, Ynm_c
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
+    .. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
+    .. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/
+
+    """
+
+    @classmethod
+    def eval(cls, n, m, theta, phi):
+        if m.is_positive:
+            zz = (Ynm(n, m, theta, phi) + Ynm_c(n, m, theta, phi)) / sqrt(2)
+            return zz
+        elif m.is_zero:
+            return Ynm(n, m, theta, phi)
+        elif m.is_negative:
+            zz = (Ynm(n, m, theta, phi) - Ynm_c(n, m, theta, phi)) / (sqrt(2)*I)
+            return zz

rl/Lib/site-packages/sympy/functions/special/tensor_functions.py

@@ -0,0 +1,474 @@
+from math import prod
+
+from sympy.core import S, Integer
+from sympy.core.function import Function
+from sympy.core.logic import fuzzy_not
+from sympy.core.relational import Ne
+from sympy.core.sorting import default_sort_key
+from sympy.external.gmpy import SYMPY_INTS
+from sympy.functions.combinatorial.factorials import factorial
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.utilities.iterables import has_dups
+
+###############################################################################
+###################### Kronecker Delta, Levi-Civita etc. ######################
+###############################################################################
+
+
+def Eijk(*args, **kwargs):
+    """
+    Represent the Levi-Civita symbol.
+
+    This is a compatibility wrapper to ``LeviCivita()``.
+
+    See Also
+    ========
+
+    LeviCivita
+
+    """
+    return LeviCivita(*args, **kwargs)
+
+
+def eval_levicivita(*args):
+    """Evaluate Levi-Civita symbol."""
+    n = len(args)
+    return prod(
+        prod(args[j] - args[i] for j in range(i + 1, n))
+        / factorial(i) for i in range(n))
+    # converting factorial(i) to int is slightly faster
+
+
+class LeviCivita(Function):
+    """
+    Represent the Levi-Civita symbol.
+
+    Explanation
+    ===========
+
+    For even permutations of indices it returns 1, for odd permutations -1, and
+    for everything else (a repeated index) it returns 0.
+
+    Thus it represents an alternating pseudotensor.
+
+    Examples
+    ========
+
+    >>> from sympy import LeviCivita
+    >>> from sympy.abc import i, j, k
+    >>> LeviCivita(1, 2, 3)
+    1
+    >>> LeviCivita(1, 3, 2)
+    -1
+    >>> LeviCivita(1, 2, 2)
+    0
+    >>> LeviCivita(i, j, k)
+    LeviCivita(i, j, k)
+    >>> LeviCivita(i, j, i)
+    0
+
+    See Also
+    ========
+
+    Eijk
+
+    """
+
+    is_integer = True
+
+    @classmethod
+    def eval(cls, *args):
+        if all(isinstance(a, (SYMPY_INTS, Integer)) for a in args):
+            return eval_levicivita(*args)
+        if has_dups(args):
+            return S.Zero
+
+    def doit(self, **hints):
+        return eval_levicivita(*self.args)
+
+
+class KroneckerDelta(Function):
+    """
+    The discrete, or Kronecker, delta function.
+
+    Explanation
+    ===========
+
+    A function that takes in two integers $i$ and $j$. It returns $0$ if $i$
+    and $j$ are not equal, or it returns $1$ if $i$ and $j$ are equal.
+
+    Examples
+    ========
+
+    An example with integer indices:
+
+        >>> from sympy import KroneckerDelta
+        >>> KroneckerDelta(1, 2)
+        0
+        >>> KroneckerDelta(3, 3)
+        1
+
+    Symbolic indices:
+
+        >>> from sympy.abc import i, j, k
+        >>> KroneckerDelta(i, j)
+        KroneckerDelta(i, j)
+        >>> KroneckerDelta(i, i)
+        1
+        >>> KroneckerDelta(i, i + 1)
+        0
+        >>> KroneckerDelta(i, i + 1 + k)
+        KroneckerDelta(i, i + k + 1)
+
+    Parameters
+    ==========
+
+    i : Number, Symbol
+        The first index of the delta function.
+    j : Number, Symbol
+        The second index of the delta function.
+
+    See Also
+    ========
+
+    eval
+    DiracDelta
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Kronecker_delta
+
+    """
+
+    is_integer = True
+
+    @classmethod
+    def eval(cls, i, j, delta_range=None):
+        """
+        Evaluates the discrete delta function.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta
+        >>> from sympy.abc import i, j, k
+
+        >>> KroneckerDelta(i, j)
+        KroneckerDelta(i, j)
+        >>> KroneckerDelta(i, i)
+        1
+        >>> KroneckerDelta(i, i + 1)
+        0
+        >>> KroneckerDelta(i, i + 1 + k)
+        KroneckerDelta(i, i + k + 1)
+
+        # indirect doctest
+
+        """
+
+        if delta_range is not None:
+            dinf, dsup = delta_range
+            if (dinf - i > 0) == True:
+                return S.Zero
+            if (dinf - j > 0) == True:
+                return S.Zero
+            if (dsup - i < 0) == True:
+                return S.Zero
+            if (dsup - j < 0) == True:
+                return S.Zero
+
+        diff = i - j
+        if diff.is_zero:
+            return S.One
+        elif fuzzy_not(diff.is_zero):
+            return S.Zero
+
+        if i.assumptions0.get("below_fermi") and \
+                j.assumptions0.get("above_fermi"):
+            return S.Zero
+        if j.assumptions0.get("below_fermi") and \
+                i.assumptions0.get("above_fermi"):
+            return S.Zero
+        # to make KroneckerDelta canonical
+        # following lines will check if inputs are in order
+        # if not, will return KroneckerDelta with correct order
+        if default_sort_key(j) < default_sort_key(i):
+            if delta_range:
+                return cls(j, i, delta_range)
+            else:
+                return cls(j, i)
+
+    @property
+    def delta_range(self):
+        if len(self.args) > 2:
+            return self.args[2]
+
+    def _eval_power(self, expt):
+        if expt.is_positive:
+            return self
+        if expt.is_negative and expt is not S.NegativeOne:
+            return 1/self
+
+    @property
+    def is_above_fermi(self):
+        """
+        True if Delta can be non-zero above fermi.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> q = Symbol('q')
+        >>> KroneckerDelta(p, a).is_above_fermi
+        True
+        >>> KroneckerDelta(p, i).is_above_fermi
+        False
+        >>> KroneckerDelta(p, q).is_above_fermi
+        True
+
+        See Also
+        ========
+
+        is_below_fermi, is_only_below_fermi, is_only_above_fermi
+
+        """
+        if self.args[0].assumptions0.get("below_fermi"):
+            return False
+        if self.args[1].assumptions0.get("below_fermi"):
+            return False
+        return True
+
+    @property
+    def is_below_fermi(self):
+        """
+        True if Delta can be non-zero below fermi.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> q = Symbol('q')
+        >>> KroneckerDelta(p, a).is_below_fermi
+        False
+        >>> KroneckerDelta(p, i).is_below_fermi
+        True
+        >>> KroneckerDelta(p, q).is_below_fermi
+        True
+
+        See Also
+        ========
+
+        is_above_fermi, is_only_above_fermi, is_only_below_fermi
+
+        """
+        if self.args[0].assumptions0.get("above_fermi"):
+            return False
+        if self.args[1].assumptions0.get("above_fermi"):
+            return False
+        return True
+
+    @property
+    def is_only_above_fermi(self):
+        """
+        True if Delta is restricted to above fermi.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> q = Symbol('q')
+        >>> KroneckerDelta(p, a).is_only_above_fermi
+        True
+        >>> KroneckerDelta(p, q).is_only_above_fermi
+        False
+        >>> KroneckerDelta(p, i).is_only_above_fermi
+        False
+
+        See Also
+        ========
+
+        is_above_fermi, is_below_fermi, is_only_below_fermi
+
+        """
+        return ( self.args[0].assumptions0.get("above_fermi")
+                or
+                self.args[1].assumptions0.get("above_fermi")
+                ) or False
+
+    @property
+    def is_only_below_fermi(self):
+        """
+        True if Delta is restricted to below fermi.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> q = Symbol('q')
+        >>> KroneckerDelta(p, i).is_only_below_fermi
+        True
+        >>> KroneckerDelta(p, q).is_only_below_fermi
+        False
+        >>> KroneckerDelta(p, a).is_only_below_fermi
+        False
+
+        See Also
+        ========
+
+        is_above_fermi, is_below_fermi, is_only_above_fermi
+
+        """
+        return ( self.args[0].assumptions0.get("below_fermi")
+                or
+                self.args[1].assumptions0.get("below_fermi")
+                ) or False
+
+    @property
+    def indices_contain_equal_information(self):
+        """
+        Returns True if indices are either both above or below fermi.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> q = Symbol('q')
+        >>> KroneckerDelta(p, q).indices_contain_equal_information
+        True
+        >>> KroneckerDelta(p, q+1).indices_contain_equal_information
+        True
+        >>> KroneckerDelta(i, p).indices_contain_equal_information
+        False
+
+        """
+        if (self.args[0].assumptions0.get("below_fermi") and
+                self.args[1].assumptions0.get("below_fermi")):
+            return True
+        if (self.args[0].assumptions0.get("above_fermi")
+                and self.args[1].assumptions0.get("above_fermi")):
+            return True
+
+        # if both indices are general we are True, else false
+        return self.is_below_fermi and self.is_above_fermi
+
+    @property
+    def preferred_index(self):
+        """
+        Returns the index which is preferred to keep in the final expression.
+
+        Explanation
+        ===========
+
+        The preferred index is the index with more information regarding fermi
+        level. If indices contain the same information, 'a' is preferred before
+        'b'.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> j = Symbol('j', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> KroneckerDelta(p, i).preferred_index
+        i
+        >>> KroneckerDelta(p, a).preferred_index
+        a
+        >>> KroneckerDelta(i, j).preferred_index
+        i
+
+        See Also
+        ========
+
+        killable_index
+
+        """
+        if self._get_preferred_index():
+            return self.args[1]
+        else:
+            return self.args[0]
+
+    @property
+    def killable_index(self):
+        """
+        Returns the index which is preferred to substitute in the final
+        expression.
+
+        Explanation
+        ===========
+
+        The index to substitute is the index with less information regarding
+        fermi level. If indices contain the same information, 'a' is preferred
+        before 'b'.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> j = Symbol('j', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> KroneckerDelta(p, i).killable_index
+        p
+        >>> KroneckerDelta(p, a).killable_index
+        p
+        >>> KroneckerDelta(i, j).killable_index
+        j
+
+        See Also
+        ========
+
+        preferred_index
+
+        """
+        if self._get_preferred_index():
+            return self.args[0]
+        else:
+            return self.args[1]
+
+    def _get_preferred_index(self):
+        """
+        Returns the index which is preferred to keep in the final expression.
+
+        The preferred index is the index with more information regarding fermi
+        level. If indices contain the same information, index 0 is returned.
+
+        """
+        if not self.is_above_fermi:
+            if self.args[0].assumptions0.get("below_fermi"):
+                return 0
+            else:
+                return 1
+        elif not self.is_below_fermi:
+            if self.args[0].assumptions0.get("above_fermi"):
+                return 0
+            else:
+                return 1
+        else:
+            return 0
+
+    @property
+    def indices(self):
+        return self.args[0:2]
+
+    def _eval_rewrite_as_Piecewise(self, *args, **kwargs):
+        i, j = args
+        return Piecewise((0, Ne(i, j)), (1, True))

rl/Lib/site-packages/sympy/functions/special/tests/test_bessel.py

@@ -0,0 +1,765 @@
+from itertools import product
+
+from sympy.concrete.summations import Sum
+from sympy.core.function import (diff, expand_func)
+from sympy.core.numbers import (I, Rational, oo, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.complexes import (conjugate, polar_lift)
+from sympy.functions.elementary.exponential import (exp, exp_polar, log)
+from sympy.functions.elementary.hyperbolic import (cosh, sinh)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.functions.special.bessel import (besseli, besselj, besselk, bessely, hankel1, hankel2, hn1, hn2, jn, jn_zeros, yn)
+from sympy.functions.special.gamma_functions import (gamma, uppergamma)
+from sympy.functions.special.hyper import hyper
+from sympy.integrals.integrals import Integral
+from sympy.series.order import O
+from sympy.series.series import series
+from sympy.functions.special.bessel import (airyai, airybi,
+                                            airyaiprime, airybiprime, marcumq)
+from sympy.core.random import (random_complex_number as randcplx,
+                                      verify_numerically as tn,
+                                      test_derivative_numerically as td,
+                                      _randint)
+from sympy.simplify import besselsimp
+from sympy.testing.pytest import raises, slow
+
+from sympy.abc import z, n, k, x
+
+randint = _randint()
+
+
+def test_bessel_rand():
+    for f in [besselj, bessely, besseli, besselk, hankel1, hankel2]:
+        assert td(f(randcplx(), z), z)
+
+    for f in [jn, yn, hn1, hn2]:
+        assert td(f(randint(-10, 10), z), z)
+
+
+def test_bessel_twoinputs():
+    for f in [besselj, bessely, besseli, besselk, hankel1, hankel2, jn, yn]:
+        raises(TypeError, lambda: f(1))
+        raises(TypeError, lambda: f(1, 2, 3))
+
+
+def test_besselj_leading_term():
+    assert besselj(0, x).as_leading_term(x) == 1
+    assert besselj(1, sin(x)).as_leading_term(x) == x/2
+    assert besselj(1, 2*sqrt(x)).as_leading_term(x) == sqrt(x)
+
+    # https://github.com/sympy/sympy/issues/21701
+    assert (besselj(z, x)/x**z).as_leading_term(x) == 1/(2**z*gamma(z + 1))
+
+
+def test_bessely_leading_term():
+    assert bessely(0, x).as_leading_term(x) == (2*log(x) - 2*log(2) + 2*S.EulerGamma)/pi
+    assert bessely(1, sin(x)).as_leading_term(x) == -2/(pi*x)
+    assert bessely(1, 2*sqrt(x)).as_leading_term(x) == -1/(pi*sqrt(x))
+
+
+def test_besseli_leading_term():
+    assert besseli(0, x).as_leading_term(x) == 1
+    assert besseli(1, sin(x)).as_leading_term(x) == x/2
+    assert besseli(1, 2*sqrt(x)).as_leading_term(x) == sqrt(x)
+
+
+def test_besselk_leading_term():
+    assert besselk(0, x).as_leading_term(x) == -log(x) - S.EulerGamma + log(2)
+    assert besselk(1, sin(x)).as_leading_term(x) == 1/x
+    assert besselk(1, 2*sqrt(x)).as_leading_term(x) == 1/(2*sqrt(x))
+
+
+def test_besselj_series():
+    assert besselj(0, x).series(x) == 1 - x**2/4 + x**4/64 + O(x**6)
+    assert besselj(0, x**(1.1)).series(x) == 1 + x**4.4/64 - x**2.2/4 + O(x**6)
+    assert besselj(0, x**2 + x).series(x) == 1 - x**2/4 - x**3/2\
+        - 15*x**4/64 + x**5/16 + O(x**6)
+    assert besselj(0, sqrt(x) + x).series(x, n=4) == 1 - x/4 - 15*x**2/64\
+        + 215*x**3/2304 - x**Rational(3, 2)/2 + x**Rational(5, 2)/16\
+        + 23*x**Rational(7, 2)/384 + O(x**4)
+    assert besselj(0, x/(1 - x)).series(x) == 1 - x**2/4 - x**3/2 - 47*x**4/64\
+        - 15*x**5/16 + O(x**6)
+    assert besselj(0, log(1 + x)).series(x) == 1 - x**2/4 + x**3/4\
+        - 41*x**4/192 + 17*x**5/96 + O(x**6)
+    assert besselj(1, sin(x)).series(x) == x/2 - 7*x**3/48 + 73*x**5/1920 + O(x**6)
+    assert besselj(1, 2*sqrt(x)).series(x) == sqrt(x) - x**Rational(3, 2)/2\
+        + x**Rational(5, 2)/12 - x**Rational(7, 2)/144 + x**Rational(9, 2)/2880\
+        - x**Rational(11, 2)/86400 + O(x**6)
+    assert besselj(-2, sin(x)).series(x, n=4) == besselj(2, sin(x)).series(x, n=4)
+
+
+def test_bessely_series():
+    const = 2*S.EulerGamma/pi - 2*log(2)/pi + 2*log(x)/pi
+    assert bessely(0, x).series(x, n=4) == const + x**2*(-log(x)/(2*pi)\
+        + (2 - 2*S.EulerGamma)/(4*pi) + log(2)/(2*pi)) + O(x**4*log(x))
+    assert bessely(1, x).series(x, n=4) == -2/(pi*x) + x*(log(x)/pi - log(2)/pi - \
+        (1 - 2*S.EulerGamma)/(2*pi)) + x**3*(-log(x)/(8*pi) + \
+        (S(5)/2 - 2*S.EulerGamma)/(16*pi) + log(2)/(8*pi)) + O(x**4*log(x))
+    assert bessely(2, x).series(x, n=4) == -4/(pi*x**2) - 1/pi + x**2*(log(x)/(4*pi) - \
+        log(2)/(4*pi) - (S(3)/2 - 2*S.EulerGamma)/(8*pi)) + O(x**4*log(x))
+    assert bessely(3, x).series(x, n=4) == -16/(pi*x**3) - 2/(pi*x) - \
+        x/(4*pi) + x**3*(log(x)/(24*pi) - log(2)/(24*pi) - \
+        (S(11)/6 - 2*S.EulerGamma)/(48*pi)) + O(x**4*log(x))
+    assert bessely(0, x**(1.1)).series(x, n=4) == 2*S.EulerGamma/pi\
+        - 2*log(2)/pi + 2.2*log(x)/pi + x**2.2*(-0.55*log(x)/pi\
+        + (2 - 2*S.EulerGamma)/(4*pi) + log(2)/(2*pi)) + O(x**4*log(x))
+    assert bessely(0, x**2 + x).series(x, n=4) == \
+        const - (2 - 2*S.EulerGamma)*(-x**3/(2*pi) - x**2/(4*pi)) + 2*x/pi\
+        + x**2*(-log(x)/(2*pi) - 1/pi + log(2)/(2*pi))\
+        + x**3*(-log(x)/pi + 1/(6*pi) + log(2)/pi) + O(x**4*log(x))
+    assert bessely(0, x/(1 - x)).series(x, n=3) == const\
+        + 2*x/pi + x**2*(-log(x)/(2*pi) + (2 - 2*S.EulerGamma)/(4*pi)\
+        + log(2)/(2*pi) + 1/pi) + O(x**3*log(x))
+    assert bessely(0, log(1 + x)).series(x, n=3) == const\
+        - x/pi + x**2*(-log(x)/(2*pi) + (2 - 2*S.EulerGamma)/(4*pi)\
+        + log(2)/(2*pi) + 5/(12*pi)) + O(x**3*log(x))
+    assert bessely(1, sin(x)).series(x, n=4) == -1/(pi*(-x**3/12 + x/2)) - \
+        (1 - 2*S.EulerGamma)*(-x**3/12 + x/2)/pi + x*(log(x)/pi - log(2)/pi) + \
+        x**3*(-7*log(x)/(24*pi) - 1/(6*pi) + (S(5)/2 - 2*S.EulerGamma)/(16*pi) +
+        7*log(2)/(24*pi)) + O(x**4*log(x))
+    assert bessely(1, 2*sqrt(x)).series(x, n=3) == -1/(pi*sqrt(x)) + \
+        sqrt(x)*(log(x)/pi - (1 - 2*S.EulerGamma)/pi) + x**(S(3)/2)*(-log(x)/(2*pi) + \
+        (S(5)/2 - 2*S.EulerGamma)/(2*pi)) + x**(S(5)/2)*(log(x)/(12*pi) - \
+        (S(10)/3 - 2*S.EulerGamma)/(12*pi)) + O(x**3*log(x))
+    assert bessely(-2, sin(x)).series(x, n=4) == bessely(2, sin(x)).series(x, n=4)
+
+
+def test_besseli_series():
+    assert besseli(0, x).series(x) == 1 + x**2/4 + x**4/64 + O(x**6)
+    assert besseli(0, x**(1.1)).series(x) == 1 + x**4.4/64 + x**2.2/4 + O(x**6)
+    assert besseli(0, x**2 + x).series(x) == 1 + x**2/4 + x**3/2 + 17*x**4/64 + \
+        x**5/16 + O(x**6)
+    assert besseli(0, sqrt(x) + x).series(x, n=4) == 1 + x/4 + 17*x**2/64 + \
+        217*x**3/2304 + x**(S(3)/2)/2 + x**(S(5)/2)/16 + 25*x**(S(7)/2)/384 + O(x**4)
+    assert besseli(0, x/(1 - x)).series(x) == 1 + x**2/4 + x**3/2 + 49*x**4/64 + \
+        17*x**5/16 + O(x**6)
+    assert besseli(0, log(1 + x)).series(x) == 1 + x**2/4 - x**3/4 + 47*x**4/192 - \
+        23*x**5/96 + O(x**6)
+    assert besseli(1, sin(x)).series(x) == x/2 - x**3/48 - 47*x**5/1920 + O(x**6)
+    assert besseli(1, 2*sqrt(x)).series(x) == sqrt(x) + x**(S(3)/2)/2 + x**(S(5)/2)/12 + \
+        x**(S(7)/2)/144 + x**(S(9)/2)/2880 + x**(S(11)/2)/86400 + O(x**6)
+    assert besseli(-2, sin(x)).series(x, n=4) == besseli(2, sin(x)).series(x, n=4)
+
+
+def test_besselk_series():
+    const = log(2) - S.EulerGamma - log(x)
+    assert besselk(0, x).series(x, n=4) == const + \
+        x**2*(-log(x)/4 - S.EulerGamma/4 + log(2)/4 + S(1)/4) + O(x**4*log(x))
+    assert besselk(1, x).series(x, n=4) == 1/x + x*(log(x)/2 - log(2)/2 - \
+        S(1)/4 + S.EulerGamma/2) + x**3*(log(x)/16 - S(5)/64 - log(2)/16 + \
+        S.EulerGamma/16) + O(x**4*log(x))
+    assert besselk(2, x).series(x, n=4) == 2/x**2 - S(1)/2 + x**2*(-log(x)/8 - \
+        S.EulerGamma/8 + log(2)/8 + S(3)/32) + O(x**4*log(x))
+    assert besselk(0, x**(1.1)).series(x, n=4) == log(2) - S.EulerGamma - \
+        1.1*log(x) + x**2.2*(-0.275*log(x) - S.EulerGamma/4 + \
+        log(2)/4 + S(1)/4) + O(x**4*log(x))
+    assert besselk(0, x**2 + x).series(x, n=4) == const + \
+        (2 - 2*S.EulerGamma)*(x**3/4 + x**2/8) - x + x**2*(-log(x)/4 + \
+        log(2)/4 + S(1)/2) + x**3*(-log(x)/2 - S(7)/12 + log(2)/2) + O(x**4*log(x))
+    assert besselk(0, x/(1 - x)).series(x, n=3) == const - x + x**2*(-log(x)/4 - \
+        S(1)/4 - S.EulerGamma/4 + log(2)/4) + O(x**3*log(x))
+    assert besselk(0, log(1 + x)).series(x, n=3) == const + x/2 + \
+        x**2*(-log(x)/4 - S.EulerGamma/4 + S(1)/24 + log(2)/4) + O(x**3*log(x))
+    assert besselk(1, 2*sqrt(x)).series(x, n=3) == 1/(2*sqrt(x)) + \
+        sqrt(x)*(log(x)/2 - S(1)/2 + S.EulerGamma) + x**(S(3)/2)*(log(x)/4 - S(5)/8 + \
+        S.EulerGamma/2) + x**(S(5)/2)*(log(x)/24 - S(5)/36 + S.EulerGamma/12) + O(x**3*log(x))
+    assert besselk(-2, sin(x)).series(x, n=4) == besselk(2, sin(x)).series(x, n=4)
+
+
+def test_diff():
+    assert besselj(n, z).diff(z) == besselj(n - 1, z)/2 - besselj(n + 1, z)/2
+    assert bessely(n, z).diff(z) == bessely(n - 1, z)/2 - bessely(n + 1, z)/2
+    assert besseli(n, z).diff(z) == besseli(n - 1, z)/2 + besseli(n + 1, z)/2
+    assert besselk(n, z).diff(z) == -besselk(n - 1, z)/2 - besselk(n + 1, z)/2
+    assert hankel1(n, z).diff(z) == hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2
+    assert hankel2(n, z).diff(z) == hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2
+
+
+def test_rewrite():
+    assert besselj(n, z).rewrite(jn) == sqrt(2*z/pi)*jn(n - S.Half, z)
+    assert bessely(n, z).rewrite(yn) == sqrt(2*z/pi)*yn(n - S.Half, z)
+    assert besseli(n, z).rewrite(besselj) == \
+        exp(-I*n*pi/2)*besselj(n, polar_lift(I)*z)
+    assert besselj(n, z).rewrite(besseli) == \
+        exp(I*n*pi/2)*besseli(n, polar_lift(-I)*z)
+
+    nu = randcplx()
+
+    assert tn(besselj(nu, z), besselj(nu, z).rewrite(besseli), z)
+    assert tn(besselj(nu, z), besselj(nu, z).rewrite(bessely), z)
+
+    assert tn(besseli(nu, z), besseli(nu, z).rewrite(besselj), z)
+    assert tn(besseli(nu, z), besseli(nu, z).rewrite(bessely), z)
+
+    assert tn(bessely(nu, z), bessely(nu, z).rewrite(besselj), z)
+    assert tn(bessely(nu, z), bessely(nu, z).rewrite(besseli), z)
+
+    assert tn(besselk(nu, z), besselk(nu, z).rewrite(besselj), z)
+    assert tn(besselk(nu, z), besselk(nu, z).rewrite(besseli), z)
+    assert tn(besselk(nu, z), besselk(nu, z).rewrite(bessely), z)
+
+    # check that a rewrite was triggered, when the order is set to a generic
+    # symbol 'nu'
+    assert yn(nu, z) != yn(nu, z).rewrite(jn)
+    assert hn1(nu, z) != hn1(nu, z).rewrite(jn)
+    assert hn2(nu, z) != hn2(nu, z).rewrite(jn)
+    assert jn(nu, z) != jn(nu, z).rewrite(yn)
+    assert hn1(nu, z) != hn1(nu, z).rewrite(yn)
+    assert hn2(nu, z) != hn2(nu, z).rewrite(yn)
+
+    # rewriting spherical bessel functions (SBFs) w.r.t. besselj, bessely is
+    # not allowed if a generic symbol 'nu' is used as the order of the SBFs
+    # to avoid inconsistencies (the order of bessel[jy] is allowed to be
+    # complex-valued, whereas SBFs are defined only for integer orders)
+    order = nu
+    for f in (besselj, bessely):
+        assert hn1(order, z) == hn1(order, z).rewrite(f)
+        assert hn2(order, z) == hn2(order, z).rewrite(f)
+
+    assert jn(order, z).rewrite(besselj) == sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(order + S.Half, z)/2
+    assert jn(order, z).rewrite(bessely) == (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-order - S.Half, z)/2
+
+    # for integral orders rewriting SBFs w.r.t bessel[jy] is allowed
+    N = Symbol('n', integer=True)
+    ri = randint(-11, 10)
+    for order in (ri, N):
+        for f in (besselj, bessely):
+            assert yn(order, z) != yn(order, z).rewrite(f)
+            assert jn(order, z) != jn(order, z).rewrite(f)
+            assert hn1(order, z) != hn1(order, z).rewrite(f)
+            assert hn2(order, z) != hn2(order, z).rewrite(f)
+
+    for func, refunc in product((yn, jn, hn1, hn2),
+                                (jn, yn, besselj, bessely)):
+        assert tn(func(ri, z), func(ri, z).rewrite(refunc), z)
+
+
+def test_expand():
+    assert expand_func(besselj(S.Half, z).rewrite(jn)) == \
+        sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
+    assert expand_func(bessely(S.Half, z).rewrite(yn)) == \
+        -sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
+
+    # XXX: teach sin/cos to work around arguments like
+    # x*exp_polar(I*pi*n/2).  Then change besselsimp -> expand_func
+    assert besselsimp(besselj(S.Half, z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
+    assert besselsimp(besselj(Rational(-1, 2), z)) == sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
+    assert besselsimp(besselj(Rational(5, 2), z)) == \
+        -sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2))
+    assert besselsimp(besselj(Rational(-5, 2), z)) == \
+        -sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2))
+
+    assert besselsimp(bessely(S.Half, z)) == \
+        -(sqrt(2)*cos(z))/(sqrt(pi)*sqrt(z))
+    assert besselsimp(bessely(Rational(-1, 2), z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
+    assert besselsimp(bessely(Rational(5, 2), z)) == \
+        sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2))
+    assert besselsimp(bessely(Rational(-5, 2), z)) == \
+        -sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2))
+
+    assert besselsimp(besseli(S.Half, z)) == sqrt(2)*sinh(z)/(sqrt(pi)*sqrt(z))
+    assert besselsimp(besseli(Rational(-1, 2), z)) == \
+        sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
+    assert besselsimp(besseli(Rational(5, 2), z)) == \
+        sqrt(2)*(z**2*sinh(z) - 3*z*cosh(z) + 3*sinh(z))/(sqrt(pi)*z**Rational(5, 2))
+    assert besselsimp(besseli(Rational(-5, 2), z)) == \
+        sqrt(2)*(z**2*cosh(z) - 3*z*sinh(z) + 3*cosh(z))/(sqrt(pi)*z**Rational(5, 2))
+
+    assert besselsimp(besselk(S.Half, z)) == \
+        besselsimp(besselk(Rational(-1, 2), z)) == sqrt(pi)*exp(-z)/(sqrt(2)*sqrt(z))
+    assert besselsimp(besselk(Rational(5, 2), z)) == \
+        besselsimp(besselk(Rational(-5, 2), z)) == \
+        sqrt(2)*sqrt(pi)*(z**2 + 3*z + 3)*exp(-z)/(2*z**Rational(5, 2))
+
+    n = Symbol('n', integer=True, positive=True)
+
+    assert expand_func(besseli(n + 2, z)) == \
+        besseli(n, z) + (-2*n - 2)*(-2*n*besseli(n, z)/z + besseli(n - 1, z))/z
+    assert expand_func(besselj(n + 2, z)) == \
+        -besselj(n, z) + (2*n + 2)*(2*n*besselj(n, z)/z - besselj(n - 1, z))/z
+    assert expand_func(besselk(n + 2, z)) == \
+        besselk(n, z) + (2*n + 2)*(2*n*besselk(n, z)/z + besselk(n - 1, z))/z
+    assert expand_func(bessely(n + 2, z)) == \
+        -bessely(n, z) + (2*n + 2)*(2*n*bessely(n, z)/z - bessely(n - 1, z))/z
+
+    assert expand_func(besseli(n + S.Half, z).rewrite(jn)) == \
+        (sqrt(2)*sqrt(z)*exp(-I*pi*(n + S.Half)/2) *
+         exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi))
+    assert expand_func(besselj(n + S.Half, z).rewrite(jn)) == \
+        sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi)
+
+    r = Symbol('r', real=True)
+    p = Symbol('p', positive=True)
+    i = Symbol('i', integer=True)
+
+    for besselx in [besselj, bessely, besseli, besselk]:
+        assert besselx(i, p).is_extended_real is True
+        assert besselx(i, x).is_extended_real is None
+        assert besselx(x, z).is_extended_real is None
+
+    for besselx in [besselj, besseli]:
+        assert besselx(i, r).is_extended_real is True
+    for besselx in [bessely, besselk]:
+        assert besselx(i, r).is_extended_real is None
+
+    for besselx in [besselj, bessely, besseli, besselk]:
+        assert expand_func(besselx(oo, x)) == besselx(oo, x, evaluate=False)
+        assert expand_func(besselx(-oo, x)) == besselx(-oo, x, evaluate=False)
+
+
+# Quite varying time, but often really slow
+@slow
+def test_slow_expand():
+    def check(eq, ans):
+        return tn(eq, ans) and eq == ans
+
+    rn = randcplx(a=1, b=0, d=0, c=2)
+
+    for besselx in [besselj, bessely, besseli, besselk]:
+        ri = S(2*randint(-11, 10) + 1) / 2  # half integer in [-21/2, 21/2]
+        assert tn(besselsimp(besselx(ri, z)), besselx(ri, z))
+
+    assert check(expand_func(besseli(rn, x)),
+                 besseli(rn - 2, x) - 2*(rn - 1)*besseli(rn - 1, x)/x)
+    assert check(expand_func(besseli(-rn, x)),
+                 besseli(-rn + 2, x) + 2*(-rn + 1)*besseli(-rn + 1, x)/x)
+
+    assert check(expand_func(besselj(rn, x)),
+                 -besselj(rn - 2, x) + 2*(rn - 1)*besselj(rn - 1, x)/x)
+    assert check(expand_func(besselj(-rn, x)),
+                 -besselj(-rn + 2, x) + 2*(-rn + 1)*besselj(-rn + 1, x)/x)
+
+    assert check(expand_func(besselk(rn, x)),
+                 besselk(rn - 2, x) + 2*(rn - 1)*besselk(rn - 1, x)/x)
+    assert check(expand_func(besselk(-rn, x)),
+                 besselk(-rn + 2, x) - 2*(-rn + 1)*besselk(-rn + 1, x)/x)
+
+    assert check(expand_func(bessely(rn, x)),
+                 -bessely(rn - 2, x) + 2*(rn - 1)*bessely(rn - 1, x)/x)
+    assert check(expand_func(bessely(-rn, x)),
+                 -bessely(-rn + 2, x) + 2*(-rn + 1)*bessely(-rn + 1, x)/x)
+
+
+def mjn(n, z):
+    return expand_func(jn(n, z))
+
+
+def myn(n, z):
+    return expand_func(yn(n, z))
+
+
+def test_jn():
+    z = symbols("z")
+    assert jn(0, 0) == 1
+    assert jn(1, 0) == 0
+    assert jn(-1, 0) == S.ComplexInfinity
+    assert jn(z, 0) == jn(z, 0, evaluate=False)
+    assert jn(0, oo) == 0
+    assert jn(0, -oo) == 0
+
+    assert mjn(0, z) == sin(z)/z
+    assert mjn(1, z) == sin(z)/z**2 - cos(z)/z
+    assert mjn(2, z) == (3/z**3 - 1/z)*sin(z) - (3/z**2) * cos(z)
+    assert mjn(3, z) == (15/z**4 - 6/z**2)*sin(z) + (1/z - 15/z**3)*cos(z)
+    assert mjn(4, z) == (1/z + 105/z**5 - 45/z**3)*sin(z) + \
+        (-105/z**4 + 10/z**2)*cos(z)
+    assert mjn(5, z) == (945/z**6 - 420/z**4 + 15/z**2)*sin(z) + \
+        (-1/z - 945/z**5 + 105/z**3)*cos(z)
+    assert mjn(6, z) == (-1/z + 10395/z**7 - 4725/z**5 + 210/z**3)*sin(z) + \
+        (-10395/z**6 + 1260/z**4 - 21/z**2)*cos(z)
+
+    assert expand_func(jn(n, z)) == jn(n, z)
+
+    # SBFs not defined for complex-valued orders
+    assert jn(2+3j, 5.2+0.3j).evalf() == jn(2+3j, 5.2+0.3j)
+
+    assert eq([jn(2, 5.2+0.3j).evalf(10)],
+              [0.09941975672 - 0.05452508024*I])
+
+
+def test_yn():
+    z = symbols("z")
+    assert myn(0, z) == -cos(z)/z
+    assert myn(1, z) == -cos(z)/z**2 - sin(z)/z
+    assert myn(2, z) == -((3/z**3 - 1/z)*cos(z) + (3/z**2)*sin(z))
+    assert expand_func(yn(n, z)) == yn(n, z)
+
+    # SBFs not defined for complex-valued orders
+    assert yn(2+3j, 5.2+0.3j).evalf() == yn(2+3j, 5.2+0.3j)
+
+    assert eq([yn(2, 5.2+0.3j).evalf(10)],
+              [0.185250342 + 0.01489557397*I])
+
+
+def test_sympify_yn():
+    assert S(15) in myn(3, pi).atoms()
+    assert myn(3, pi) == 15/pi**4 - 6/pi**2
+
+
+def eq(a, b, tol=1e-6):
+    for u, v in zip(a, b):
+        if not (abs(u - v) < tol):
+            return False
+    return True
+
+
+def test_jn_zeros():
+    assert eq(jn_zeros(0, 4), [3.141592, 6.283185, 9.424777, 12.566370])
+    assert eq(jn_zeros(1, 4), [4.493409, 7.725251, 10.904121, 14.066193])
+    assert eq(jn_zeros(2, 4), [5.763459, 9.095011, 12.322940, 15.514603])
+    assert eq(jn_zeros(3, 4), [6.987932, 10.417118, 13.698023, 16.923621])
+    assert eq(jn_zeros(4, 4), [8.182561, 11.704907, 15.039664, 18.301255])
+
+
+def test_bessel_eval():
+    n, m, k = Symbol('n', integer=True), Symbol('m'), Symbol('k', integer=True, zero=False)
+
+    for f in [besselj, besseli]:
+        assert f(0, 0) is S.One
+        assert f(2.1, 0) is S.Zero
+        assert f(-3, 0) is S.Zero
+        assert f(-10.2, 0) is S.ComplexInfinity
+        assert f(1 + 3*I, 0) is S.Zero
+        assert f(-3 + I, 0) is S.ComplexInfinity
+        assert f(-2*I, 0) is S.NaN
+        assert f(n, 0) != S.One and f(n, 0) != S.Zero
+        assert f(m, 0) != S.One and f(m, 0) != S.Zero
+        assert f(k, 0) is S.Zero
+
+    assert bessely(0, 0) is S.NegativeInfinity
+    assert besselk(0, 0) is S.Infinity
+    for f in [bessely, besselk]:
+        assert f(1 + I, 0) is S.ComplexInfinity
+        assert f(I, 0) is S.NaN
+
+    for f in [besselj, bessely]:
+        assert f(m, S.Infinity) is S.Zero
+        assert f(m, S.NegativeInfinity) is S.Zero
+
+    for f in [besseli, besselk]:
+        assert f(m, I*S.Infinity) is S.Zero
+        assert f(m, I*S.NegativeInfinity) is S.Zero
+
+    for f in [besseli, besselk]:
+        assert f(-4, z) == f(4, z)
+        assert f(-3, z) == f(3, z)
+        assert f(-n, z) == f(n, z)
+        assert f(-m, z) != f(m, z)
+
+    for f in [besselj, bessely]:
+        assert f(-4, z) == f(4, z)
+        assert f(-3, z) == -f(3, z)
+        assert f(-n, z) == (-1)**n*f(n, z)
+        assert f(-m, z) != (-1)**m*f(m, z)
+
+    for f in [besselj, besseli]:
+        assert f(m, -z) == (-z)**m*z**(-m)*f(m, z)
+
+    assert besseli(2, -z) == besseli(2, z)
+    assert besseli(3, -z) == -besseli(3, z)
+
+    assert besselj(0, -z) == besselj(0, z)
+    assert besselj(1, -z) == -besselj(1, z)
+
+    assert besseli(0, I*z) == besselj(0, z)
+    assert besseli(1, I*z) == I*besselj(1, z)
+    assert besselj(3, I*z) == -I*besseli(3, z)
+
+
+def test_bessel_nan():
+    # FIXME: could have these return NaN; for now just fix infinite recursion
+    for f in [besselj, bessely, besseli, besselk, hankel1, hankel2, yn, jn]:
+        assert f(1, S.NaN) == f(1, S.NaN, evaluate=False)
+
+
+def test_meromorphic():
+    assert besselj(2, x).is_meromorphic(x, 1) == True
+    assert besselj(2, x).is_meromorphic(x, 0) == True
+    assert besselj(2, x).is_meromorphic(x, oo) == False
+    assert besselj(S(2)/3, x).is_meromorphic(x, 1) == True
+    assert besselj(S(2)/3, x).is_meromorphic(x, 0) == False
+    assert besselj(S(2)/3, x).is_meromorphic(x, oo) == False
+    assert besselj(x, 2*x).is_meromorphic(x, 2) == False
+    assert besselk(0, x).is_meromorphic(x, 1) == True
+    assert besselk(2, x).is_meromorphic(x, 0) == True
+    assert besseli(0, x).is_meromorphic(x, 1) == True
+    assert besseli(2, x).is_meromorphic(x, 0) == True
+    assert bessely(0, x).is_meromorphic(x, 1) == True
+    assert bessely(0, x).is_meromorphic(x, 0) == False
+    assert bessely(2, x).is_meromorphic(x, 0) == True
+    assert hankel1(3, x**2 + 2*x).is_meromorphic(x, 1) == True
+    assert hankel1(0, x).is_meromorphic(x, 0) == False
+    assert hankel2(11, 4).is_meromorphic(x, 5) == True
+    assert hn1(6, 7*x**3 + 4).is_meromorphic(x, 7) == True
+    assert hn2(3, 2*x).is_meromorphic(x, 9) == True
+    assert jn(5, 2*x + 7).is_meromorphic(x, 4) == True
+    assert yn(8, x**2 + 11).is_meromorphic(x, 6) == True
+
+
+def test_conjugate():
+    n = Symbol('n')
+    z = Symbol('z', extended_real=False)
+    x = Symbol('x', extended_real=True)
+    y = Symbol('y', positive=True)
+    t = Symbol('t', negative=True)
+
+    for f in [besseli, besselj, besselk, bessely, hankel1, hankel2]:
+        assert f(n, -1).conjugate() != f(conjugate(n), -1)
+        assert f(n, x).conjugate() != f(conjugate(n), x)
+        assert f(n, t).conjugate() != f(conjugate(n), t)
+
+    rz = randcplx(b=0.5)
+
+    for f in [besseli, besselj, besselk, bessely]:
+        assert f(n, 1 + I).conjugate() == f(conjugate(n), 1 - I)
+        assert f(n, 0).conjugate() == f(conjugate(n), 0)
+        assert f(n, 1).conjugate() == f(conjugate(n), 1)
+        assert f(n, z).conjugate() == f(conjugate(n), conjugate(z))
+        assert f(n, y).conjugate() == f(conjugate(n), y)
+        assert tn(f(n, rz).conjugate(), f(conjugate(n), conjugate(rz)))
+
+    assert hankel1(n, 1 + I).conjugate() == hankel2(conjugate(n), 1 - I)
+    assert hankel1(n, 0).conjugate() == hankel2(conjugate(n), 0)
+    assert hankel1(n, 1).conjugate() == hankel2(conjugate(n), 1)
+    assert hankel1(n, y).conjugate() == hankel2(conjugate(n), y)
+    assert hankel1(n, z).conjugate() == hankel2(conjugate(n), conjugate(z))
+    assert tn(hankel1(n, rz).conjugate(), hankel2(conjugate(n), conjugate(rz)))
+
+    assert hankel2(n, 1 + I).conjugate() == hankel1(conjugate(n), 1 - I)
+    assert hankel2(n, 0).conjugate() == hankel1(conjugate(n), 0)
+    assert hankel2(n, 1).conjugate() == hankel1(conjugate(n), 1)
+    assert hankel2(n, y).conjugate() == hankel1(conjugate(n), y)
+    assert hankel2(n, z).conjugate() == hankel1(conjugate(n), conjugate(z))
+    assert tn(hankel2(n, rz).conjugate(), hankel1(conjugate(n), conjugate(rz)))
+
+
+def test_branching():
+    assert besselj(polar_lift(k), x) == besselj(k, x)
+    assert besseli(polar_lift(k), x) == besseli(k, x)
+
+    n = Symbol('n', integer=True)
+    assert besselj(n, exp_polar(2*pi*I)*x) == besselj(n, x)
+    assert besselj(n, polar_lift(x)) == besselj(n, x)
+    assert besseli(n, exp_polar(2*pi*I)*x) == besseli(n, x)
+    assert besseli(n, polar_lift(x)) == besseli(n, x)
+
+    def tn(func, s):
+        from sympy.core.random import uniform
+        c = uniform(1, 5)
+        expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
+        eps = 1e-15
+        expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
+        return abs(expr.n() - expr2.n()).n() < 1e-10
+
+    nu = Symbol('nu')
+    assert besselj(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besselj(nu, x)
+    assert besseli(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besseli(nu, x)
+    assert tn(besselj, 2)
+    assert tn(besselj, pi)
+    assert tn(besselj, I)
+    assert tn(besseli, 2)
+    assert tn(besseli, pi)
+    assert tn(besseli, I)
+
+
+def test_airy_base():
+    z = Symbol('z')
+    x = Symbol('x', real=True)
+    y = Symbol('y', real=True)
+
+    assert conjugate(airyai(z)) == airyai(conjugate(z))
+    assert airyai(x).is_extended_real
+
+    assert airyai(x+I*y).as_real_imag() == (
+            airyai(x - I*y)/2 + airyai(x + I*y)/2,
+            I*(airyai(x - I*y) - airyai(x + I*y))/2)
+
+
+def test_airyai():
+    z = Symbol('z', real=False)
+    t = Symbol('t', negative=True)
+    p = Symbol('p', positive=True)
+
+    assert isinstance(airyai(z), airyai)
+
+    assert airyai(0) == 3**Rational(1, 3)/(3*gamma(Rational(2, 3)))
+    assert airyai(oo) == 0
+    assert airyai(-oo) == 0
+
+    assert diff(airyai(z), z) == airyaiprime(z)
+
+    assert series(airyai(z), z, 0, 3) == (
+        3**Rational(5, 6)*gamma(Rational(1, 3))/(6*pi) - 3**Rational(1, 6)*z*gamma(Rational(2, 3))/(2*pi) + O(z**3))
+
+    assert airyai(z).rewrite(hyper) == (
+        -3**Rational(2, 3)*z*hyper((), (Rational(4, 3),), z**3/9)/(3*gamma(Rational(1, 3))) +
+         3**Rational(1, 3)*hyper((), (Rational(2, 3),), z**3/9)/(3*gamma(Rational(2, 3))))
+
+    assert isinstance(airyai(z).rewrite(besselj), airyai)
+    assert airyai(t).rewrite(besselj) == (
+        sqrt(-t)*(besselj(Rational(-1, 3), 2*(-t)**Rational(3, 2)/3) +
+                  besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3)
+    assert airyai(z).rewrite(besseli) == (
+        -z*besseli(Rational(1, 3), 2*z**Rational(3, 2)/3)/(3*(z**Rational(3, 2))**Rational(1, 3)) +
+         (z**Rational(3, 2))**Rational(1, 3)*besseli(Rational(-1, 3), 2*z**Rational(3, 2)/3)/3)
+    assert airyai(p).rewrite(besseli) == (
+        sqrt(p)*(besseli(Rational(-1, 3), 2*p**Rational(3, 2)/3) -
+                 besseli(Rational(1, 3), 2*p**Rational(3, 2)/3))/3)
+
+    assert expand_func(airyai(2*(3*z**5)**Rational(1, 3))) == (
+        -sqrt(3)*(-1 + (z**5)**Rational(1, 3)/z**Rational(5, 3))*airybi(2*3**Rational(1, 3)*z**Rational(5, 3))/6 +
+         (1 + (z**5)**Rational(1, 3)/z**Rational(5, 3))*airyai(2*3**Rational(1, 3)*z**Rational(5, 3))/2)
+
+
+def test_airybi():
+    z = Symbol('z', real=False)
+    t = Symbol('t', negative=True)
+    p = Symbol('p', positive=True)
+
+    assert isinstance(airybi(z), airybi)
+
+    assert airybi(0) == 3**Rational(5, 6)/(3*gamma(Rational(2, 3)))
+    assert airybi(oo) is oo
+    assert airybi(-oo) == 0
+
+    assert diff(airybi(z), z) == airybiprime(z)
+
+    assert series(airybi(z), z, 0, 3) == (
+        3**Rational(1, 3)*gamma(Rational(1, 3))/(2*pi) + 3**Rational(2, 3)*z*gamma(Rational(2, 3))/(2*pi) + O(z**3))
+
+    assert airybi(z).rewrite(hyper) == (
+        3**Rational(1, 6)*z*hyper((), (Rational(4, 3),), z**3/9)/gamma(Rational(1, 3)) +
+        3**Rational(5, 6)*hyper((), (Rational(2, 3),), z**3/9)/(3*gamma(Rational(2, 3))))
+
+    assert isinstance(airybi(z).rewrite(besselj), airybi)
+    assert airyai(t).rewrite(besselj) == (
+        sqrt(-t)*(besselj(Rational(-1, 3), 2*(-t)**Rational(3, 2)/3) +
+                  besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3)
+    assert airybi(z).rewrite(besseli) == (
+        sqrt(3)*(z*besseli(Rational(1, 3), 2*z**Rational(3, 2)/3)/(z**Rational(3, 2))**Rational(1, 3) +
+                 (z**Rational(3, 2))**Rational(1, 3)*besseli(Rational(-1, 3), 2*z**Rational(3, 2)/3))/3)
+    assert airybi(p).rewrite(besseli) == (
+        sqrt(3)*sqrt(p)*(besseli(Rational(-1, 3), 2*p**Rational(3, 2)/3) +
+                         besseli(Rational(1, 3), 2*p**Rational(3, 2)/3))/3)
+
+    assert expand_func(airybi(2*(3*z**5)**Rational(1, 3))) == (
+        sqrt(3)*(1 - (z**5)**Rational(1, 3)/z**Rational(5, 3))*airyai(2*3**Rational(1, 3)*z**Rational(5, 3))/2 +
+        (1 + (z**5)**Rational(1, 3)/z**Rational(5, 3))*airybi(2*3**Rational(1, 3)*z**Rational(5, 3))/2)
+
+
+def test_airyaiprime():
+    z = Symbol('z', real=False)
+    t = Symbol('t', negative=True)
+    p = Symbol('p', positive=True)
+
+    assert isinstance(airyaiprime(z), airyaiprime)
+
+    assert airyaiprime(0) == -3**Rational(2, 3)/(3*gamma(Rational(1, 3)))
+    assert airyaiprime(oo) == 0
+
+    assert diff(airyaiprime(z), z) == z*airyai(z)
+
+    assert series(airyaiprime(z), z, 0, 3) == (
+        -3**Rational(2, 3)/(3*gamma(Rational(1, 3))) + 3**Rational(1, 3)*z**2/(6*gamma(Rational(2, 3))) + O(z**3))
+
+    assert airyaiprime(z).rewrite(hyper) == (
+        3**Rational(1, 3)*z**2*hyper((), (Rational(5, 3),), z**3/9)/(6*gamma(Rational(2, 3))) -
+        3**Rational(2, 3)*hyper((), (Rational(1, 3),), z**3/9)/(3*gamma(Rational(1, 3))))
+
+    assert isinstance(airyaiprime(z).rewrite(besselj), airyaiprime)
+    assert airyai(t).rewrite(besselj) == (
+        sqrt(-t)*(besselj(Rational(-1, 3), 2*(-t)**Rational(3, 2)/3) +
+                  besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3)
+    assert airyaiprime(z).rewrite(besseli) == (
+        z**2*besseli(Rational(2, 3), 2*z**Rational(3, 2)/3)/(3*(z**Rational(3, 2))**Rational(2, 3)) -
+        (z**Rational(3, 2))**Rational(2, 3)*besseli(Rational(-1, 3), 2*z**Rational(3, 2)/3)/3)
+    assert airyaiprime(p).rewrite(besseli) == (
+        p*(-besseli(Rational(-2, 3), 2*p**Rational(3, 2)/3) + besseli(Rational(2, 3), 2*p**Rational(3, 2)/3))/3)
+
+    assert expand_func(airyaiprime(2*(3*z**5)**Rational(1, 3))) == (
+        sqrt(3)*(z**Rational(5, 3)/(z**5)**Rational(1, 3) - 1)*airybiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/6 +
+        (z**Rational(5, 3)/(z**5)**Rational(1, 3) + 1)*airyaiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/2)
+
+
+def test_airybiprime():
+    z = Symbol('z', real=False)
+    t = Symbol('t', negative=True)
+    p = Symbol('p', positive=True)
+
+    assert isinstance(airybiprime(z), airybiprime)
+
+    assert airybiprime(0) == 3**Rational(1, 6)/gamma(Rational(1, 3))
+    assert airybiprime(oo) is oo
+    assert airybiprime(-oo) == 0
+
+    assert diff(airybiprime(z), z) == z*airybi(z)
+
+    assert series(airybiprime(z), z, 0, 3) == (
+        3**Rational(1, 6)/gamma(Rational(1, 3)) + 3**Rational(5, 6)*z**2/(6*gamma(Rational(2, 3))) + O(z**3))
+
+    assert airybiprime(z).rewrite(hyper) == (
+        3**Rational(5, 6)*z**2*hyper((), (Rational(5, 3),), z**3/9)/(6*gamma(Rational(2, 3))) +
+        3**Rational(1, 6)*hyper((), (Rational(1, 3),), z**3/9)/gamma(Rational(1, 3)))
+
+    assert isinstance(airybiprime(z).rewrite(besselj), airybiprime)
+    assert airyai(t).rewrite(besselj) == (
+        sqrt(-t)*(besselj(Rational(-1, 3), 2*(-t)**Rational(3, 2)/3) +
+                  besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3)
+    assert airybiprime(z).rewrite(besseli) == (
+        sqrt(3)*(z**2*besseli(Rational(2, 3), 2*z**Rational(3, 2)/3)/(z**Rational(3, 2))**Rational(2, 3) +
+                 (z**Rational(3, 2))**Rational(2, 3)*besseli(Rational(-2, 3), 2*z**Rational(3, 2)/3))/3)
+    assert airybiprime(p).rewrite(besseli) == (
+        sqrt(3)*p*(besseli(Rational(-2, 3), 2*p**Rational(3, 2)/3) + besseli(Rational(2, 3), 2*p**Rational(3, 2)/3))/3)
+
+    assert expand_func(airybiprime(2*(3*z**5)**Rational(1, 3))) == (
+        sqrt(3)*(z**Rational(5, 3)/(z**5)**Rational(1, 3) - 1)*airyaiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/2 +
+        (z**Rational(5, 3)/(z**5)**Rational(1, 3) + 1)*airybiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/2)
+
+
+def test_marcumq():
+    m = Symbol('m')
+    a = Symbol('a')
+    b = Symbol('b')
+
+    assert marcumq(0, 0, 0) == 0
+    assert marcumq(m, 0, b) == uppergamma(m, b**2/2)/gamma(m)
+    assert marcumq(2, 0, 5) == 27*exp(Rational(-25, 2))/2
+    assert marcumq(0, a, 0) == 1 - exp(-a**2/2)
+    assert marcumq(0, pi, 0) == 1 - exp(-pi**2/2)
+    assert marcumq(1, a, a) == S.Half + exp(-a**2)*besseli(0, a**2)/2
+    assert marcumq(2, a, a) == S.Half + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2)
+
+    assert diff(marcumq(1, a, 3), a) == a*(-marcumq(1, a, 3) + marcumq(2, a, 3))
+    assert diff(marcumq(2, 3, b), b) == -b**2*exp(-b**2/2 - Rational(9, 2))*besseli(1, 3*b)/3
+
+    x = Symbol('x')
+    assert marcumq(2, 3, 4).rewrite(Integral, x=x) == \
+           Integral(x**2*exp(-x**2/2 - Rational(9, 2))*besseli(1, 3*x), (x, 4, oo))/3
+    assert eq([marcumq(5, -2, 3).rewrite(Integral).evalf(10)],
+              [0.7905769565])
+
+    k = Symbol('k')
+    assert marcumq(-3, -5, -7).rewrite(Sum, k=k) == \
+           exp(-37)*Sum((Rational(5, 7))**k*besseli(k, 35), (k, 4, oo))
+    assert eq([marcumq(1, 3, 1).rewrite(Sum).evalf(10)],
+              [0.9891705502])
+
+    assert marcumq(1, a, a, evaluate=False).rewrite(besseli) == S.Half + exp(-a**2)*besseli(0, a**2)/2
+    assert marcumq(2, a, a, evaluate=False).rewrite(besseli) == S.Half + exp(-a**2)*besseli(0, a**2)/2 + \
+           exp(-a**2)*besseli(1, a**2)
+    assert marcumq(3, a, a).rewrite(besseli) == (besseli(1, a**2) + besseli(2, a**2))*exp(-a**2) + \
+           S.Half + exp(-a**2)*besseli(0, a**2)/2
+    assert marcumq(5, 8, 8).rewrite(besseli) == exp(-64)*besseli(0, 64)/2 + \
+           (besseli(4, 64) + besseli(3, 64) + besseli(2, 64) + besseli(1, 64))*exp(-64) + S.Half
+    assert marcumq(m, a, a).rewrite(besseli) == marcumq(m, a, a)
+
+    x = Symbol('x', integer=True)
+    assert marcumq(x, a, a).rewrite(besseli) == marcumq(x, a, a)
+
+
+def test_issue_26134():
+    x = Symbol('x')
+    assert marcumq(2, 3, 4).rewrite(Integral, x=x).dummy_eq(
+        Integral(x**2*exp(-x**2/2 - Rational(9, 2))*besseli(1, 3*x), (x, 4, oo))/3)

rl/Lib/site-packages/sympy/functions/special/tests/test_beta_functions.py

@@ -0,0 +1,89 @@
+from sympy.core.function import (diff, expand_func)
+from sympy.core.numbers import I, Rational, pi
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, symbols)
+from sympy.functions.combinatorial.numbers import catalan
+from sympy.functions.elementary.complexes import conjugate
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.special.beta_functions import (beta, betainc, betainc_regularized)
+from sympy.functions.special.gamma_functions import gamma, polygamma
+from sympy.functions.special.hyper import hyper
+from sympy.integrals.integrals import Integral
+from sympy.core.function import ArgumentIndexError
+from sympy.core.expr import unchanged
+from sympy.testing.pytest import raises
+
+
+def test_beta():
+    x, y = symbols('x y')
+    t = Dummy('t')
+
+    assert unchanged(beta, x, y)
+    assert unchanged(beta, x, x)
+
+    assert beta(5, -3).is_real == True
+    assert beta(3, y).is_real is None
+
+    assert expand_func(beta(x, y)) == gamma(x)*gamma(y)/gamma(x + y)
+    assert expand_func(beta(x, y) - beta(y, x)) == 0  # Symmetric
+    assert expand_func(beta(x, y)) == expand_func(beta(x, y + 1) + beta(x + 1, y)).simplify()
+
+    assert diff(beta(x, y), x) == beta(x, y)*(polygamma(0, x) - polygamma(0, x + y))
+    assert diff(beta(x, y), y) == beta(x, y)*(polygamma(0, y) - polygamma(0, x + y))
+
+    assert conjugate(beta(x, y)) == beta(conjugate(x), conjugate(y))
+
+    raises(ArgumentIndexError, lambda: beta(x, y).fdiff(3))
+
+    assert beta(x, y).rewrite(gamma) == gamma(x)*gamma(y)/gamma(x + y)
+    assert beta(x).rewrite(gamma) == gamma(x)**2/gamma(2*x)
+    assert beta(x, y).rewrite(Integral).dummy_eq(Integral(t**(x - 1) * (1 - t)**(y - 1), (t, 0, 1)))
+    assert beta(Rational(-19, 10), Rational(-1, 10)) == S.Zero
+    assert beta(Rational(-19, 10), Rational(-9, 10)) == \
+        800*2**(S(4)/5)*sqrt(pi)*gamma(S.One/10)/(171*gamma(-S(7)/5))
+    assert beta(Rational(19, 10), Rational(29, 10)) == 100/(551*catalan(Rational(19, 10)))
+    assert beta(1, 0) == S.ComplexInfinity
+    assert beta(0, 1) == S.ComplexInfinity
+    assert beta(2, 3) == S.One/12
+    assert unchanged(beta, x, x + 1)
+    assert unchanged(beta, x, 1)
+    assert unchanged(beta, 1, y)
+    assert beta(x, x + 1).doit() == 1/(x*(x+1)*catalan(x))
+    assert beta(1, y).doit() == 1/y
+    assert beta(x, 1).doit() == 1/x
+    assert beta(Rational(-19, 10), Rational(-1, 10), evaluate=False).doit() == S.Zero
+    assert beta(2) == beta(2, 2)
+    assert beta(x, evaluate=False) != beta(x, x)
+    assert beta(x, evaluate=False).doit() == beta(x, x)
+
+
+def test_betainc():
+    a, b, x1, x2 = symbols('a b x1 x2')
+
+    assert unchanged(betainc, a, b, x1, x2)
+    assert unchanged(betainc, a, b, 0, x1)
+
+    assert betainc(1, 2, 0, -5).is_real == True
+    assert betainc(1, 2, 0, x2).is_real is None
+    assert conjugate(betainc(I, 2, 3 - I, 1 + 4*I)) == betainc(-I, 2, 3 + I, 1 - 4*I)
+
+    assert betainc(a, b, 0, 1).rewrite(Integral).dummy_eq(beta(a, b).rewrite(Integral))
+    assert betainc(1, 2, 0, x2).rewrite(hyper) == x2*hyper((1, -1), (2,), x2)
+
+    assert betainc(1, 2, 3, 3).evalf() == 0
+
+
+def test_betainc_regularized():
+    a, b, x1, x2 = symbols('a b x1 x2')
+
+    assert unchanged(betainc_regularized, a, b, x1, x2)
+    assert unchanged(betainc_regularized, a, b, 0, x1)
+
+    assert betainc_regularized(3, 5, 0, -1).is_real == True
+    assert betainc_regularized(3, 5, 0, x2).is_real is None
+    assert conjugate(betainc_regularized(3*I, 1, 2 + I, 1 + 2*I)) == betainc_regularized(-3*I, 1, 2 - I, 1 - 2*I)
+
+    assert betainc_regularized(a, b, 0, 1).rewrite(Integral) == 1
+    assert betainc_regularized(1, 2, x1, x2).rewrite(hyper) == 2*x2*hyper((1, -1), (2,), x2) - 2*x1*hyper((1, -1), (2,), x1)
+
+    assert betainc_regularized(4, 1, 5, 5).evalf() == 0

rl/Lib/site-packages/sympy/functions/special/tests/test_bsplines.py

@@ -0,0 +1,167 @@
+from sympy.functions import bspline_basis_set, interpolating_spline
+from sympy.core.numbers import Rational
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.logic.boolalg import And
+from sympy.sets.sets import Interval
+from sympy.testing.pytest import slow
+
+x, y = symbols('x,y')
+
+
+def test_basic_degree_0():
+    d = 0
+    knots = range(5)
+    splines = bspline_basis_set(d, knots, x)
+    for i in range(len(splines)):
+        assert splines[i] == Piecewise((1, Interval(i, i + 1).contains(x)),
+                                       (0, True))
+
+
+def test_basic_degree_1():
+    d = 1
+    knots = range(5)
+    splines = bspline_basis_set(d, knots, x)
+    assert splines[0] == Piecewise((x, Interval(0, 1).contains(x)),
+                                   (2 - x, Interval(1, 2).contains(x)),
+                                   (0, True))
+    assert splines[1] == Piecewise((-1 + x, Interval(1, 2).contains(x)),
+                                   (3 - x, Interval(2, 3).contains(x)),
+                                   (0, True))
+    assert splines[2] == Piecewise((-2 + x, Interval(2, 3).contains(x)),
+                                   (4 - x, Interval(3, 4).contains(x)),
+                                   (0, True))
+
+
+def test_basic_degree_2():
+    d = 2
+    knots = range(5)
+    splines = bspline_basis_set(d, knots, x)
+    b0 = Piecewise((x**2/2, Interval(0, 1).contains(x)),
+                   (Rational(-3, 2) + 3*x - x**2, Interval(1, 2).contains(x)),
+                   (Rational(9, 2) - 3*x + x**2/2, Interval(2, 3).contains(x)),
+                   (0, True))
+    b1 = Piecewise((S.Half - x + x**2/2, Interval(1, 2).contains(x)),
+                   (Rational(-11, 2) + 5*x - x**2, Interval(2, 3).contains(x)),
+                   (8 - 4*x + x**2/2, Interval(3, 4).contains(x)),
+                   (0, True))
+    assert splines[0] == b0
+    assert splines[1] == b1
+
+
+def test_basic_degree_3():
+    d = 3
+    knots = range(5)
+    splines = bspline_basis_set(d, knots, x)
+    b0 = Piecewise(
+        (x**3/6, Interval(0, 1).contains(x)),
+        (Rational(2, 3) - 2*x + 2*x**2 - x**3/2, Interval(1, 2).contains(x)),
+        (Rational(-22, 3) + 10*x - 4*x**2 + x**3/2, Interval(2, 3).contains(x)),
+        (Rational(32, 3) - 8*x + 2*x**2 - x**3/6, Interval(3, 4).contains(x)),
+        (0, True)
+    )
+    assert splines[0] == b0
+
+
+def test_repeated_degree_1():
+    d = 1
+    knots = [0, 0, 1, 2, 2, 3, 4, 4]
+    splines = bspline_basis_set(d, knots, x)
+    assert splines[0] == Piecewise((1 - x, Interval(0, 1).contains(x)),
+                                   (0, True))
+    assert splines[1] == Piecewise((x, Interval(0, 1).contains(x)),
+                                   (2 - x, Interval(1, 2).contains(x)),
+                                   (0, True))
+    assert splines[2] == Piecewise((-1 + x, Interval(1, 2).contains(x)),
+                                   (0, True))
+    assert splines[3] == Piecewise((3 - x, Interval(2, 3).contains(x)),
+                                   (0, True))
+    assert splines[4] == Piecewise((-2 + x, Interval(2, 3).contains(x)),
+                                   (4 - x, Interval(3, 4).contains(x)),
+                                   (0, True))
+    assert splines[5] == Piecewise((-3 + x, Interval(3, 4).contains(x)),
+                                   (0, True))
+
+
+def test_repeated_degree_2():
+    d = 2
+    knots = [0, 0, 1, 2, 2, 3, 4, 4]
+    splines = bspline_basis_set(d, knots, x)
+
+    assert splines[0] == Piecewise(((-3*x**2/2 + 2*x), And(x <= 1, x >= 0)),
+                                   (x**2/2 - 2*x + 2, And(x <= 2, x >= 1)),
+                                   (0, True))
+    assert splines[1] == Piecewise((x**2/2, And(x <= 1, x >= 0)),
+                                   (-3*x**2/2 + 4*x - 2, And(x <= 2, x >= 1)),
+                                   (0, True))
+    assert splines[2] == Piecewise((x**2 - 2*x + 1, And(x <= 2, x >= 1)),
+                                   (x**2 - 6*x + 9, And(x <= 3, x >= 2)),
+                                   (0, True))
+    assert splines[3] == Piecewise((-3*x**2/2 + 8*x - 10, And(x <= 3, x >= 2)),
+                                   (x**2/2 - 4*x + 8, And(x <= 4, x >= 3)),
+                                   (0, True))
+    assert splines[4] == Piecewise((x**2/2 - 2*x + 2, And(x <= 3, x >= 2)),
+                                   (-3*x**2/2 + 10*x - 16, And(x <= 4, x >= 3)),
+                                   (0, True))
+
+# Tests for interpolating_spline
+
+
+def test_10_points_degree_1():
+    d = 1
+    X = [-5, 2, 3, 4, 7, 9, 10, 30, 31, 34]
+    Y = [-10, -2, 2, 4, 7, 6, 20, 45, 19, 25]
+    spline = interpolating_spline(d, x, X, Y)
+
+    assert spline == Piecewise((x*Rational(8, 7) - Rational(30, 7), (x >= -5) & (x <= 2)), (4*x - 10, (x >= 2) & (x <= 3)),
+                               (2*x - 4, (x >= 3) & (x <= 4)), (x, (x >= 4) & (x <= 7)),
+                               (-x/2 + Rational(21, 2), (x >= 7) & (x <= 9)), (14*x - 120, (x >= 9) & (x <= 10)),
+                               (x*Rational(5, 4) + Rational(15, 2), (x >= 10) & (x <= 30)), (-26*x + 825, (x >= 30) & (x <= 31)),
+                               (2*x - 43, (x >= 31) & (x <= 34)))
+
+
+def test_3_points_degree_2():
+    d = 2
+    X = [-3, 10, 19]
+    Y = [3, -4, 30]
+    spline = interpolating_spline(d, x, X, Y)
+
+    assert spline == Piecewise((505*x**2/2574 - x*Rational(4921, 2574) - Rational(1931, 429), (x >= -3) & (x <= 19)))
+
+
+def test_5_points_degree_2():
+    d = 2
+    X = [-3, 2, 4, 5, 10]
+    Y = [-1, 2, 5, 10, 14]
+    spline = interpolating_spline(d, x, X, Y)
+
+    assert spline == Piecewise((4*x**2/329 + x*Rational(1007, 1645) + Rational(1196, 1645), (x >= -3) & (x <= 3)),
+                               (2701*x**2/1645 - x*Rational(15079, 1645) + Rational(5065, 329), (x >= 3) & (x <= Rational(9, 2))),
+                               (-1319*x**2/1645 + x*Rational(21101, 1645) - Rational(11216, 329), (x >= Rational(9, 2)) & (x <= 10)))
+
+
+@slow
+def test_6_points_degree_3():
+    d = 3
+    X = [-1, 0, 2, 3, 9, 12]
+    Y = [-4, 3, 3, 7, 9, 20]
+    spline = interpolating_spline(d, x, X, Y)
+
+    assert spline == Piecewise((6058*x**3/5301 - 18427*x**2/5301 + x*Rational(12622, 5301) + 3, (x >= -1) & (x <= 2)),
+                               (-8327*x**3/5301 + 67883*x**2/5301 - x*Rational(159998, 5301) + Rational(43661, 1767), (x >= 2) & (x <= 3)),
+                               (5414*x**3/47709 - 1386*x**2/589 + x*Rational(4267, 279) - Rational(12232, 589), (x >= 3) & (x <= 12)))
+
+
+def test_issue_19262():
+    Delta = symbols('Delta', positive=True)
+    knots = [i*Delta for i in range(4)]
+    basis = bspline_basis_set(1, knots, x)
+    y = symbols('y', nonnegative=True)
+    basis2 = bspline_basis_set(1, knots, y)
+    assert basis[0].subs(x, y) == basis2[0]
+    assert interpolating_spline(1, x,
+        [Delta*i for i in [1, 2, 4, 7]], [3, 6, 5, 7]
+        )  == Piecewise((3*x/Delta, (Delta <= x) & (x <= 2*Delta)),
+        (7 - x/(2*Delta), (x >= 2*Delta) & (x <= 4*Delta)),
+        (Rational(7, 3) + 2*x/(3*Delta), (x >= 4*Delta) & (x <= 7*Delta)))

rl/Lib/site-packages/sympy/functions/special/tests/test_delta_functions.py

@@ -0,0 +1,165 @@
+from sympy.core.numbers import (I, nan, oo, pi)
+from sympy.core.relational import (Eq, Ne)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.complexes import (adjoint, conjugate, sign, transpose)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.special.delta_functions import (DiracDelta, Heaviside)
+from sympy.functions.special.singularity_functions import SingularityFunction
+from sympy.simplify.simplify import signsimp
+
+
+from sympy.testing.pytest import raises
+
+from sympy.core.expr import unchanged
+
+from sympy.core.function import ArgumentIndexError
+
+
+x, y = symbols('x y')
+i = symbols('t', nonzero=True)
+j = symbols('j', positive=True)
+k = symbols('k', negative=True)
+
+def test_DiracDelta():
+    assert DiracDelta(1) == 0
+    assert DiracDelta(5.1) == 0
+    assert DiracDelta(-pi) == 0
+    assert DiracDelta(5, 7) == 0
+    assert DiracDelta(x, 0) == DiracDelta(x)
+    assert DiracDelta(i) == 0
+    assert DiracDelta(j) == 0
+    assert DiracDelta(k) == 0
+    assert DiracDelta(nan) is nan
+    assert DiracDelta(0).func is DiracDelta
+    assert DiracDelta(x).func is DiracDelta
+    # FIXME: this is generally undefined @ x=0
+    #         But then limit(Delta(c)*Heaviside(x),x,-oo)
+    #         need's to be implemented.
+    # assert 0*DiracDelta(x) == 0
+
+    assert adjoint(DiracDelta(x)) == DiracDelta(x)
+    assert adjoint(DiracDelta(x - y)) == DiracDelta(x - y)
+    assert conjugate(DiracDelta(x)) == DiracDelta(x)
+    assert conjugate(DiracDelta(x - y)) == DiracDelta(x - y)
+    assert transpose(DiracDelta(x)) == DiracDelta(x)
+    assert transpose(DiracDelta(x - y)) == DiracDelta(x - y)
+
+    assert DiracDelta(x).diff(x) == DiracDelta(x, 1)
+    assert DiracDelta(x, 1).diff(x) == DiracDelta(x, 2)
+
+    assert DiracDelta(x).is_simple(x) is True
+    assert DiracDelta(3*x).is_simple(x) is True
+    assert DiracDelta(x**2).is_simple(x) is False
+    assert DiracDelta(sqrt(x)).is_simple(x) is False
+    assert DiracDelta(x).is_simple(y) is False
+
+    assert DiracDelta(x*y).expand(diracdelta=True, wrt=x) == DiracDelta(x)/abs(y)
+    assert DiracDelta(x*y).expand(diracdelta=True, wrt=y) == DiracDelta(y)/abs(x)
+    assert DiracDelta(x**2*y).expand(diracdelta=True, wrt=x) == DiracDelta(x**2*y)
+    assert DiracDelta(y).expand(diracdelta=True, wrt=x) == DiracDelta(y)
+    assert DiracDelta((x - 1)*(x - 2)*(x - 3)).expand(diracdelta=True, wrt=x) == (
+        DiracDelta(x - 3)/2 + DiracDelta(x - 2) + DiracDelta(x - 1)/2)
+
+    assert DiracDelta(2*x) != DiracDelta(x)  # scaling property
+    assert DiracDelta(x) == DiracDelta(-x)  # even function
+    assert DiracDelta(-x, 2) == DiracDelta(x, 2)
+    assert DiracDelta(-x, 1) == -DiracDelta(x, 1)  # odd deriv is odd
+    assert DiracDelta(-oo*x) == DiracDelta(oo*x)
+    assert DiracDelta(x - y) != DiracDelta(y - x)
+    assert signsimp(DiracDelta(x - y) - DiracDelta(y - x)) == 0
+
+    assert DiracDelta(x*y).expand(diracdelta=True, wrt=x) == DiracDelta(x)/abs(y)
+    assert DiracDelta(x*y).expand(diracdelta=True, wrt=y) == DiracDelta(y)/abs(x)
+    assert DiracDelta(x**2*y).expand(diracdelta=True, wrt=x) == DiracDelta(x**2*y)
+    assert DiracDelta(y).expand(diracdelta=True, wrt=x) == DiracDelta(y)
+    assert DiracDelta((x - 1)*(x - 2)*(x - 3)).expand(diracdelta=True) == (
+            DiracDelta(x - 3)/2 + DiracDelta(x - 2) + DiracDelta(x - 1)/2)
+
+    raises(ArgumentIndexError, lambda: DiracDelta(x).fdiff(2))
+    raises(ValueError, lambda: DiracDelta(x, -1))
+    raises(ValueError, lambda: DiracDelta(I))
+    raises(ValueError, lambda: DiracDelta(2 + 3*I))
+
+
+def test_heaviside():
+    assert Heaviside(-5) == 0
+    assert Heaviside(1) == 1
+    assert Heaviside(0) == S.Half
+
+    assert Heaviside(0, x) == x
+    assert unchanged(Heaviside,x, nan)
+    assert Heaviside(0, nan) == nan
+
+    h0  = Heaviside(x, 0)
+    h12 = Heaviside(x, S.Half)
+    h1  = Heaviside(x, 1)
+
+    assert h0.args == h0.pargs == (x, 0)
+    assert h1.args == h1.pargs == (x, 1)
+    assert h12.args == (x, S.Half)
+    assert h12.pargs == (x,) # default 1/2 suppressed
+
+    assert adjoint(Heaviside(x)) == Heaviside(x)
+    assert adjoint(Heaviside(x - y)) == Heaviside(x - y)
+    assert conjugate(Heaviside(x)) == Heaviside(x)
+    assert conjugate(Heaviside(x - y)) == Heaviside(x - y)
+    assert transpose(Heaviside(x)) == Heaviside(x)
+    assert transpose(Heaviside(x - y)) == Heaviside(x - y)
+
+    assert Heaviside(x).diff(x) == DiracDelta(x)
+    assert Heaviside(x + I).is_Function is True
+    assert Heaviside(I*x).is_Function is True
+
+    raises(ArgumentIndexError, lambda: Heaviside(x).fdiff(2))
+    raises(ValueError, lambda: Heaviside(I))
+    raises(ValueError, lambda: Heaviside(2 + 3*I))
+
+
+def test_rewrite():
+    x, y = Symbol('x', real=True), Symbol('y')
+    assert Heaviside(x).rewrite(Piecewise) == (
+        Piecewise((0, x < 0), (Heaviside(0), Eq(x, 0)), (1, True)))
+    assert Heaviside(y).rewrite(Piecewise) == (
+        Piecewise((0, y < 0), (Heaviside(0), Eq(y, 0)), (1, True)))
+    assert Heaviside(x, y).rewrite(Piecewise) == (
+        Piecewise((0, x < 0), (y, Eq(x, 0)), (1, True)))
+    assert Heaviside(x, 0).rewrite(Piecewise) == (
+        Piecewise((0, x <= 0), (1, True)))
+    assert Heaviside(x, 1).rewrite(Piecewise) == (
+        Piecewise((0, x < 0), (1, True)))
+    assert Heaviside(x, nan).rewrite(Piecewise) == (
+        Piecewise((0, x < 0), (nan, Eq(x, 0)), (1, True)))
+
+    assert Heaviside(x).rewrite(sign) == \
+        Heaviside(x, H0=Heaviside(0)).rewrite(sign) == \
+        Piecewise(
+            (sign(x)/2 + S(1)/2, Eq(Heaviside(0), S(1)/2)),
+            (Piecewise(
+                (sign(x)/2 + S(1)/2, Ne(x, 0)), (Heaviside(0), True)), True)
+        )
+
+    assert Heaviside(y).rewrite(sign) == Heaviside(y)
+    assert Heaviside(x, S.Half).rewrite(sign) == (sign(x)+1)/2
+    assert Heaviside(x, y).rewrite(sign) == \
+        Piecewise(
+            (sign(x)/2 + S(1)/2, Eq(y, S(1)/2)),
+            (Piecewise(
+                (sign(x)/2 + S(1)/2, Ne(x, 0)), (y, True)), True)
+        )
+
+    assert DiracDelta(y).rewrite(Piecewise) == Piecewise((DiracDelta(0), Eq(y, 0)), (0, True))
+    assert DiracDelta(y, 1).rewrite(Piecewise) == DiracDelta(y, 1)
+    assert DiracDelta(x - 5).rewrite(Piecewise) == (
+        Piecewise((DiracDelta(0), Eq(x - 5, 0)), (0, True)))
+
+    assert (x*DiracDelta(x - 10)).rewrite(SingularityFunction) == x*SingularityFunction(x, 10, -1)
+    assert 5*x*y*DiracDelta(y, 1).rewrite(SingularityFunction) == 5*x*y*SingularityFunction(y, 0, -2)
+    assert DiracDelta(0).rewrite(SingularityFunction) == SingularityFunction(0, 0, -1)
+    assert DiracDelta(0, 1).rewrite(SingularityFunction) == SingularityFunction(0, 0, -2)
+
+    assert Heaviside(x).rewrite(SingularityFunction) == SingularityFunction(x, 0, 0)
+    assert 5*x*y*Heaviside(y + 1).rewrite(SingularityFunction) == 5*x*y*SingularityFunction(y, -1, 0)
+    assert ((x - 3)**3*Heaviside(x - 3)).rewrite(SingularityFunction) == (x - 3)**3*SingularityFunction(x, 3, 0)
+    assert Heaviside(0).rewrite(SingularityFunction) == S.Half

rl/Lib/site-packages/sympy/functions/special/tests/test_elliptic_integrals.py

@@ -0,0 +1,181 @@
+from sympy.core.numbers import (I, Rational, oo, pi, zoo)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol)
+from sympy.functions.elementary.hyperbolic import atanh
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (sin, tan)
+from sympy.functions.special.gamma_functions import gamma
+from sympy.functions.special.hyper import (hyper, meijerg)
+from sympy.integrals.integrals import Integral
+from sympy.series.order import O
+from sympy.functions.special.elliptic_integrals import (elliptic_k as K,
+    elliptic_f as F, elliptic_e as E, elliptic_pi as P)
+from sympy.core.random import (test_derivative_numerically as td,
+                                      random_complex_number as randcplx,
+                                      verify_numerically as tn)
+from sympy.abc import z, m, n
+
+i = Symbol('i', integer=True)
+j = Symbol('k', integer=True, positive=True)
+t = Dummy('t')
+
+def test_K():
+    assert K(0) == pi/2
+    assert K(S.Half) == 8*pi**Rational(3, 2)/gamma(Rational(-1, 4))**2
+    assert K(1) is zoo
+    assert K(-1) == gamma(Rational(1, 4))**2/(4*sqrt(2*pi))
+    assert K(oo) == 0
+    assert K(-oo) == 0
+    assert K(I*oo) == 0
+    assert K(-I*oo) == 0
+    assert K(zoo) == 0
+
+    assert K(z).diff(z) == (E(z) - (1 - z)*K(z))/(2*z*(1 - z))
+    assert td(K(z), z)
+
+    zi = Symbol('z', real=False)
+    assert K(zi).conjugate() == K(zi.conjugate())
+    zr = Symbol('z', negative=True)
+    assert K(zr).conjugate() == K(zr)
+
+    assert K(z).rewrite(hyper) == \
+        (pi/2)*hyper((S.Half, S.Half), (S.One,), z)
+    assert tn(K(z), (pi/2)*hyper((S.Half, S.Half), (S.One,), z))
+    assert K(z).rewrite(meijerg) == \
+        meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -z)/2
+    assert tn(K(z), meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -z)/2)
+
+    assert K(z).series(z) == pi/2 + pi*z/8 + 9*pi*z**2/128 + \
+        25*pi*z**3/512 + 1225*pi*z**4/32768 + 3969*pi*z**5/131072 + O(z**6)
+
+    assert K(m).rewrite(Integral).dummy_eq(
+        Integral(1/sqrt(1 - m*sin(t)**2), (t, 0, pi/2)))
+
+def test_F():
+    assert F(z, 0) == z
+    assert F(0, m) == 0
+    assert F(pi*i/2, m) == i*K(m)
+    assert F(z, oo) == 0
+    assert F(z, -oo) == 0
+
+    assert F(-z, m) == -F(z, m)
+
+    assert F(z, m).diff(z) == 1/sqrt(1 - m*sin(z)**2)
+    assert F(z, m).diff(m) == E(z, m)/(2*m*(1 - m)) - F(z, m)/(2*m) - \
+        sin(2*z)/(4*(1 - m)*sqrt(1 - m*sin(z)**2))
+    r = randcplx()
+    assert td(F(z, r), z)
+    assert td(F(r, m), m)
+
+    mi = Symbol('m', real=False)
+    assert F(z, mi).conjugate() == F(z.conjugate(), mi.conjugate())
+    mr = Symbol('m', negative=True)
+    assert F(z, mr).conjugate() == F(z.conjugate(), mr)
+
+    assert F(z, m).series(z) == \
+        z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6)
+
+    assert F(z, m).rewrite(Integral).dummy_eq(
+        Integral(1/sqrt(1 - m*sin(t)**2), (t, 0, z)))
+
+def test_E():
+    assert E(z, 0) == z
+    assert E(0, m) == 0
+    assert E(i*pi/2, m) == i*E(m)
+    assert E(z, oo) is zoo
+    assert E(z, -oo) is zoo
+    assert E(0) == pi/2
+    assert E(1) == 1
+    assert E(oo) == I*oo
+    assert E(-oo) is oo
+    assert E(zoo) is zoo
+
+    assert E(-z, m) == -E(z, m)
+
+    assert E(z, m).diff(z) == sqrt(1 - m*sin(z)**2)
+    assert E(z, m).diff(m) == (E(z, m) - F(z, m))/(2*m)
+    assert E(z).diff(z) == (E(z) - K(z))/(2*z)
+    r = randcplx()
+    assert td(E(r, m), m)
+    assert td(E(z, r), z)
+    assert td(E(z), z)
+
+    mi = Symbol('m', real=False)
+    assert E(z, mi).conjugate() == E(z.conjugate(), mi.conjugate())
+    assert E(mi).conjugate() == E(mi.conjugate())
+    mr = Symbol('m', negative=True)
+    assert E(z, mr).conjugate() == E(z.conjugate(), mr)
+    assert E(mr).conjugate() == E(mr)
+
+    assert E(z).rewrite(hyper) == (pi/2)*hyper((Rational(-1, 2), S.Half), (S.One,), z)
+    assert tn(E(z), (pi/2)*hyper((Rational(-1, 2), S.Half), (S.One,), z))
+    assert E(z).rewrite(meijerg) == \
+        -meijerg(((S.Half, Rational(3, 2)), []), ((S.Zero,), (S.Zero,)), -z)/4
+    assert tn(E(z), -meijerg(((S.Half, Rational(3, 2)), []), ((S.Zero,), (S.Zero,)), -z)/4)
+
+    assert E(z, m).series(z) == \
+        z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)
+    assert E(z).series(z) == pi/2 - pi*z/8 - 3*pi*z**2/128 - \
+        5*pi*z**3/512 - 175*pi*z**4/32768 - 441*pi*z**5/131072 + O(z**6)
+    assert E(4*z/(z+1)).series(z) == \
+        pi/2 - pi*z/2 + pi*z**2/8 - 3*pi*z**3/8 - 15*pi*z**4/128 - 93*pi*z**5/128 + O(z**6)
+
+    assert E(z, m).rewrite(Integral).dummy_eq(
+        Integral(sqrt(1 - m*sin(t)**2), (t, 0, z)))
+    assert E(m).rewrite(Integral).dummy_eq(
+        Integral(sqrt(1 - m*sin(t)**2), (t, 0, pi/2)))
+
+def test_P():
+    assert P(0, z, m) == F(z, m)
+    assert P(1, z, m) == F(z, m) + \
+        (sqrt(1 - m*sin(z)**2)*tan(z) - E(z, m))/(1 - m)
+    assert P(n, i*pi/2, m) == i*P(n, m)
+    assert P(n, z, 0) == atanh(sqrt(n - 1)*tan(z))/sqrt(n - 1)
+    assert P(n, z, n) == F(z, n) - P(1, z, n) + tan(z)/sqrt(1 - n*sin(z)**2)
+    assert P(oo, z, m) == 0
+    assert P(-oo, z, m) == 0
+    assert P(n, z, oo) == 0
+    assert P(n, z, -oo) == 0
+    assert P(0, m) == K(m)
+    assert P(1, m) is zoo
+    assert P(n, 0) == pi/(2*sqrt(1 - n))
+    assert P(2, 1) is -oo
+    assert P(-1, 1) is oo
+    assert P(n, n) == E(n)/(1 - n)
+
+    assert P(n, -z, m) == -P(n, z, m)
+
+    ni, mi = Symbol('n', real=False), Symbol('m', real=False)
+    assert P(ni, z, mi).conjugate() == \
+        P(ni.conjugate(), z.conjugate(), mi.conjugate())
+    nr, mr = Symbol('n', negative=True), \
+        Symbol('m', negative=True)
+    assert P(nr, z, mr).conjugate() == P(nr, z.conjugate(), mr)
+    assert P(n, m).conjugate() == P(n.conjugate(), m.conjugate())
+
+    assert P(n, z, m).diff(n) == (E(z, m) + (m - n)*F(z, m)/n +
+        (n**2 - m)*P(n, z, m)/n - n*sqrt(1 -
+            m*sin(z)**2)*sin(2*z)/(2*(1 - n*sin(z)**2)))/(2*(m - n)*(n - 1))
+    assert P(n, z, m).diff(z) == 1/(sqrt(1 - m*sin(z)**2)*(1 - n*sin(z)**2))
+    assert P(n, z, m).diff(m) == (E(z, m)/(m - 1) + P(n, z, m) -
+        m*sin(2*z)/(2*(m - 1)*sqrt(1 - m*sin(z)**2)))/(2*(n - m))
+    assert P(n, m).diff(n) == (E(m) + (m - n)*K(m)/n +
+        (n**2 - m)*P(n, m)/n)/(2*(m - n)*(n - 1))
+    assert P(n, m).diff(m) == (E(m)/(m - 1) + P(n, m))/(2*(n - m))
+
+    # These tests fail due to
+    # https://github.com/fredrik-johansson/mpmath/issues/571#issuecomment-777201962
+    # https://github.com/sympy/sympy/issues/20933#issuecomment-777080385
+    #
+    # rx, ry = randcplx(), randcplx()
+    # assert td(P(n, rx, ry), n)
+    # assert td(P(rx, z, ry), z)
+    # assert td(P(rx, ry, m), m)
+
+    assert P(n, z, m).series(z) == z + z**3*(m/6 + n/3) + \
+        z**5*(3*m**2/40 + m*n/10 - m/30 + n**2/5 - n/15) + O(z**6)
+
+    assert P(n, z, m).rewrite(Integral).dummy_eq(
+        Integral(1/((1 - n*sin(t)**2)*sqrt(1 - m*sin(t)**2)), (t, 0, z)))
+    assert P(n, m).rewrite(Integral).dummy_eq(
+        Integral(1/((1 - n*sin(t)**2)*sqrt(1 - m*sin(t)**2)), (t, 0, pi/2)))

rl/Lib/site-packages/sympy/functions/special/tests/test_error_functions.py

@@ -0,0 +1,837 @@
+from sympy.core.function import (diff, expand, expand_func)
+from sympy.core import EulerGamma
+from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols, Dummy)
+from sympy.functions.elementary.complexes import (conjugate, im, polar_lift, re)
+from sympy.functions.elementary.exponential import (exp, exp_polar, log)
+from sympy.functions.elementary.hyperbolic import (cosh, sinh)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin, sinc)
+from sympy.functions.special.error_functions import (Chi, Ci, E1, Ei, Li, Shi, Si, erf, erf2, erf2inv, erfc, erfcinv, erfi, erfinv, expint, fresnelc, fresnels, li)
+from sympy.functions.special.gamma_functions import (gamma, uppergamma)
+from sympy.functions.special.hyper import (hyper, meijerg)
+from sympy.integrals.integrals import (Integral, integrate)
+from sympy.series.gruntz import gruntz
+from sympy.series.limits import limit
+from sympy.series.order import O
+from sympy.core.expr import unchanged
+from sympy.core.function import ArgumentIndexError
+from sympy.functions.special.error_functions import _erfs, _eis
+from sympy.testing.pytest import raises
+
+x, y, z = symbols('x,y,z')
+w = Symbol("w", real=True)
+n = Symbol("n", integer=True)
+t = Dummy('t')
+
+
+def test_erf():
+    assert erf(nan) is nan
+
+    assert erf(oo) == 1
+    assert erf(-oo) == -1
+
+    assert erf(0) is S.Zero
+
+    assert erf(I*oo) == oo*I
+    assert erf(-I*oo) == -oo*I
+
+    assert erf(-2) == -erf(2)
+    assert erf(-x*y) == -erf(x*y)
+    assert erf(-x - y) == -erf(x + y)
+
+    assert erf(erfinv(x)) == x
+    assert erf(erfcinv(x)) == 1 - x
+    assert erf(erf2inv(0, x)) == x
+    assert erf(erf2inv(0, x, evaluate=False)) == x # To cover code in erf
+    assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x
+
+    assert erf(I).is_real is False
+    assert erf(0, evaluate=False).is_real
+    assert erf(0, evaluate=False).is_zero
+
+    assert conjugate(erf(z)) == erf(conjugate(z))
+
+    assert erf(x).as_leading_term(x) == 2*x/sqrt(pi)
+    assert erf(x*y).as_leading_term(y) == 2*x*y/sqrt(pi)
+    assert (erf(x*y)/erf(y)).as_leading_term(y) == x
+    assert erf(1/x).as_leading_term(x) == S.One
+
+    assert erf(z).rewrite('uppergamma') == sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z
+    assert erf(z).rewrite('erfc') == S.One - erfc(z)
+    assert erf(z).rewrite('erfi') == -I*erfi(I*z)
+    assert erf(z).rewrite('fresnels') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
+        I*fresnels(z*(1 - I)/sqrt(pi)))
+    assert erf(z).rewrite('fresnelc') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
+        I*fresnels(z*(1 - I)/sqrt(pi)))
+    assert erf(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi)
+    assert erf(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi)
+    assert erf(z).rewrite('expint') == sqrt(z**2)/z - z*expint(S.Half, z**2)/sqrt(S.Pi)
+
+    assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \
+        2/sqrt(pi)
+    assert limit((1 - erf(z))*exp(z**2)*z, z, oo) == 1/sqrt(pi)
+    assert limit((1 - erf(x))*exp(x**2)*sqrt(pi)*x, x, oo) == 1
+    assert limit(((1 - erf(x))*exp(x**2)*sqrt(pi)*x - 1)*2*x**2, x, oo) == -1
+    assert limit(erf(x)/x, x, 0) == 2/sqrt(pi)
+    assert limit(x**(-4) - sqrt(pi)*erf(x**2) / (2*x**6), x, 0) == S(1)/3
+
+    assert erf(x).as_real_imag() == \
+        (erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2,
+         -I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2)
+
+    assert erf(x).as_real_imag(deep=False) == \
+        (erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2,
+         -I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2)
+
+    assert erf(w).as_real_imag() == (erf(w), 0)
+    assert erf(w).as_real_imag(deep=False) == (erf(w), 0)
+    # issue 13575
+    assert erf(I).as_real_imag() == (0, -I*erf(I))
+
+    raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
+
+    assert erf(x).inverse() == erfinv
+
+
+def test_erf_series():
+    assert erf(x).series(x, 0, 7) == 2*x/sqrt(pi) - \
+        2*x**3/3/sqrt(pi) + x**5/5/sqrt(pi) + O(x**7)
+
+    assert erf(x).series(x, oo) == \
+        -exp(-x**2)*(3/(4*x**5) - 1/(2*x**3) + 1/x + O(x**(-6), (x, oo)))/sqrt(pi) + 1
+    assert erf(x**2).series(x, oo, n=8) == \
+        (-1/(2*x**6) + x**(-2) + O(x**(-8), (x, oo)))*exp(-x**4)/sqrt(pi)*-1 + 1
+    assert erf(sqrt(x)).series(x, oo, n=3) == (sqrt(1/x) - (1/x)**(S(3)/2)/2\
+        + 3*(1/x)**(S(5)/2)/4 + O(x**(-3), (x, oo)))*exp(-x)/sqrt(pi)*-1 + 1
+
+
+def test_erf_evalf():
+    assert abs( erf(Float(2.0)) - 0.995322265 ) < 1E-8 # XXX
+
+
+def test__erfs():
+    assert _erfs(z).diff(z) == -2/sqrt(S.Pi) + 2*z*_erfs(z)
+
+    assert _erfs(1/z).series(z) == \
+        z/sqrt(pi) - z**3/(2*sqrt(pi)) + 3*z**5/(4*sqrt(pi)) + O(z**6)
+
+    assert expand(erf(z).rewrite('tractable').diff(z).rewrite('intractable')) \
+        == erf(z).diff(z)
+    assert _erfs(z).rewrite("intractable") == (-erf(z) + 1)*exp(z**2)
+    raises(ArgumentIndexError, lambda: _erfs(z).fdiff(2))
+
+
+def test_erfc():
+    assert erfc(nan) is nan
+
+    assert erfc(oo) is S.Zero
+    assert erfc(-oo) == 2
+
+    assert erfc(0) == 1
+
+    assert erfc(I*oo) == -oo*I
+    assert erfc(-I*oo) == oo*I
+
+    assert erfc(-x) == S(2) - erfc(x)
+    assert erfc(erfcinv(x)) == x
+
+    assert erfc(I).is_real is False
+    assert erfc(0, evaluate=False).is_real
+    assert erfc(0, evaluate=False).is_zero is False
+
+    assert erfc(erfinv(x)) == 1 - x
+
+    assert conjugate(erfc(z)) == erfc(conjugate(z))
+
+    assert erfc(x).as_leading_term(x) is S.One
+    assert erfc(1/x).as_leading_term(x) == S.Zero
+
+    assert erfc(z).rewrite('erf') == 1 - erf(z)
+    assert erfc(z).rewrite('erfi') == 1 + I*erfi(I*z)
+    assert erfc(z).rewrite('fresnels') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
+        I*fresnels(z*(1 - I)/sqrt(pi)))
+    assert erfc(z).rewrite('fresnelc') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
+        I*fresnels(z*(1 - I)/sqrt(pi)))
+    assert erfc(z).rewrite('hyper') == 1 - 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi)
+    assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi)
+    assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z
+    assert erfc(z).rewrite('expint') == S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi)
+    assert erfc(z).rewrite('tractable') == _erfs(z)*exp(-z**2)
+    assert expand_func(erf(x) + erfc(x)) is S.One
+
+    assert erfc(x).as_real_imag() == \
+        (erfc(re(x) - I*im(x))/2 + erfc(re(x) + I*im(x))/2,
+         -I*(-erfc(re(x) - I*im(x)) + erfc(re(x) + I*im(x)))/2)
+
+    assert erfc(x).as_real_imag(deep=False) == \
+        (erfc(re(x) - I*im(x))/2 + erfc(re(x) + I*im(x))/2,
+         -I*(-erfc(re(x) - I*im(x)) + erfc(re(x) + I*im(x)))/2)
+
+    assert erfc(w).as_real_imag() == (erfc(w), 0)
+    assert erfc(w).as_real_imag(deep=False) == (erfc(w), 0)
+    raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))
+
+    assert erfc(x).inverse() == erfcinv
+
+
+def test_erfc_series():
+    assert erfc(x).series(x, 0, 7) == 1 - 2*x/sqrt(pi) + \
+        2*x**3/3/sqrt(pi) - x**5/5/sqrt(pi) + O(x**7)
+
+    assert erfc(x).series(x, oo) == \
+            (3/(4*x**5) - 1/(2*x**3) + 1/x + O(x**(-6), (x, oo)))*exp(-x**2)/sqrt(pi)
+
+
+def test_erfc_evalf():
+    assert abs( erfc(Float(2.0)) - 0.00467773 ) < 1E-8 # XXX
+
+
+def test_erfi():
+    assert erfi(nan) is nan
+
+    assert erfi(oo) is S.Infinity
+    assert erfi(-oo) is S.NegativeInfinity
+
+    assert erfi(0) is S.Zero
+
+    assert erfi(I*oo) == I
+    assert erfi(-I*oo) == -I
+
+    assert erfi(-x) == -erfi(x)
+
+    assert erfi(I*erfinv(x)) == I*x
+    assert erfi(I*erfcinv(x)) == I*(1 - x)
+    assert erfi(I*erf2inv(0, x)) == I*x
+    assert erfi(I*erf2inv(0, x, evaluate=False)) == I*x # To cover code in erfi
+
+    assert erfi(I).is_real is False
+    assert erfi(0, evaluate=False).is_real
+    assert erfi(0, evaluate=False).is_zero
+
+    assert conjugate(erfi(z)) == erfi(conjugate(z))
+
+    assert erfi(x).as_leading_term(x) == 2*x/sqrt(pi)
+    assert erfi(x*y).as_leading_term(y) == 2*x*y/sqrt(pi)
+    assert (erfi(x*y)/erfi(y)).as_leading_term(y) == x
+    assert erfi(1/x).as_leading_term(x) == erfi(1/x)
+
+    assert erfi(z).rewrite('erf') == -I*erf(I*z)
+    assert erfi(z).rewrite('erfc') == I*erfc(I*z) - I
+    assert erfi(z).rewrite('fresnels') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
+        I*fresnels(z*(1 + I)/sqrt(pi)))
+    assert erfi(z).rewrite('fresnelc') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
+        I*fresnels(z*(1 + I)/sqrt(pi)))
+    assert erfi(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], z**2)/sqrt(pi)
+    assert erfi(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [Rational(-1, 2)], -z**2)/sqrt(pi)
+    assert erfi(z).rewrite('uppergamma') == (sqrt(-z**2)/z*(uppergamma(S.Half,
+        -z**2)/sqrt(S.Pi) - S.One))
+    assert erfi(z).rewrite('expint') == sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(S.Pi)
+    assert erfi(z).rewrite('tractable') == -I*(-_erfs(I*z)*exp(z**2) + 1)
+    assert expand_func(erfi(I*z)) == I*erf(z)
+
+    assert erfi(x).as_real_imag() == \
+        (erfi(re(x) - I*im(x))/2 + erfi(re(x) + I*im(x))/2,
+         -I*(-erfi(re(x) - I*im(x)) + erfi(re(x) + I*im(x)))/2)
+    assert erfi(x).as_real_imag(deep=False) == \
+        (erfi(re(x) - I*im(x))/2 + erfi(re(x) + I*im(x))/2,
+         -I*(-erfi(re(x) - I*im(x)) + erfi(re(x) + I*im(x)))/2)
+
+    assert erfi(w).as_real_imag() == (erfi(w), 0)
+    assert erfi(w).as_real_imag(deep=False) == (erfi(w), 0)
+
+    raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
+
+
+def test_erfi_series():
+    assert erfi(x).series(x, 0, 7) == 2*x/sqrt(pi) + \
+        2*x**3/3/sqrt(pi) + x**5/5/sqrt(pi) + O(x**7)
+
+    assert erfi(x).series(x, oo) == \
+        (3/(4*x**5) + 1/(2*x**3) + 1/x + O(x**(-6), (x, oo)))*exp(x**2)/sqrt(pi) - I
+
+
+def test_erfi_evalf():
+    assert abs( erfi(Float(2.0)) - 18.5648024145756 ) < 1E-13  # XXX
+
+
+def test_erf2():
+
+    assert erf2(0, 0) is S.Zero
+    assert erf2(x, x) is S.Zero
+    assert erf2(nan, 0) is nan
+
+    assert erf2(-oo,  y) ==  erf(y) + 1
+    assert erf2( oo,  y) ==  erf(y) - 1
+    assert erf2(  x, oo) ==  1 - erf(x)
+    assert erf2(  x,-oo) == -1 - erf(x)
+    assert erf2(x, erf2inv(x, y)) == y
+
+    assert erf2(-x, -y) == -erf2(x,y)
+    assert erf2(-x,  y) == erf(y) + erf(x)
+    assert erf2( x, -y) == -erf(y) - erf(x)
+    assert erf2(x, y).rewrite('fresnels') == erf(y).rewrite(fresnels)-erf(x).rewrite(fresnels)
+    assert erf2(x, y).rewrite('fresnelc') == erf(y).rewrite(fresnelc)-erf(x).rewrite(fresnelc)
+    assert erf2(x, y).rewrite('hyper') == erf(y).rewrite(hyper)-erf(x).rewrite(hyper)
+    assert erf2(x, y).rewrite('meijerg') == erf(y).rewrite(meijerg)-erf(x).rewrite(meijerg)
+    assert erf2(x, y).rewrite('uppergamma') == erf(y).rewrite(uppergamma) - erf(x).rewrite(uppergamma)
+    assert erf2(x, y).rewrite('expint') == erf(y).rewrite(expint)-erf(x).rewrite(expint)
+
+    assert erf2(I, 0).is_real is False
+    assert erf2(0, 0, evaluate=False).is_real
+    assert erf2(0, 0, evaluate=False).is_zero
+    assert erf2(x, x, evaluate=False).is_zero
+    assert erf2(x, y).is_zero is None
+
+    assert expand_func(erf(x) + erf2(x, y)) == erf(y)
+
+    assert conjugate(erf2(x, y)) == erf2(conjugate(x), conjugate(y))
+
+    assert erf2(x, y).rewrite('erf')  == erf(y) - erf(x)
+    assert erf2(x, y).rewrite('erfc') == erfc(x) - erfc(y)
+    assert erf2(x, y).rewrite('erfi') == I*(erfi(I*x) - erfi(I*y))
+
+    assert erf2(x, y).diff(x) == erf2(x, y).fdiff(1)
+    assert erf2(x, y).diff(y) == erf2(x, y).fdiff(2)
+    assert erf2(x, y).diff(x) == -2*exp(-x**2)/sqrt(pi)
+    assert erf2(x, y).diff(y) == 2*exp(-y**2)/sqrt(pi)
+    raises(ArgumentIndexError, lambda: erf2(x, y).fdiff(3))
+
+    assert erf2(x, y).is_extended_real is None
+    xr, yr = symbols('xr yr', extended_real=True)
+    assert erf2(xr, yr).is_extended_real is True
+
+
+def test_erfinv():
+    assert erfinv(0) is S.Zero
+    assert erfinv(1) is S.Infinity
+    assert erfinv(nan) is S.NaN
+    assert erfinv(-1) is S.NegativeInfinity
+
+    assert erfinv(erf(w)) == w
+    assert erfinv(erf(-w)) == -w
+
+    assert erfinv(x).diff() == sqrt(pi)*exp(erfinv(x)**2)/2
+    raises(ArgumentIndexError, lambda: erfinv(x).fdiff(2))
+
+    assert erfinv(z).rewrite('erfcinv') == erfcinv(1-z)
+    assert erfinv(z).inverse() == erf
+
+
+def test_erfinv_evalf():
+    assert abs( erfinv(Float(0.2)) - 0.179143454621292 ) < 1E-13
+
+
+def test_erfcinv():
+    assert erfcinv(1) is S.Zero
+    assert erfcinv(0) is S.Infinity
+    assert erfcinv(nan) is S.NaN
+
+    assert erfcinv(x).diff() == -sqrt(pi)*exp(erfcinv(x)**2)/2
+    raises(ArgumentIndexError, lambda: erfcinv(x).fdiff(2))
+
+    assert erfcinv(z).rewrite('erfinv') == erfinv(1-z)
+    assert erfcinv(z).inverse() == erfc
+
+
+def test_erf2inv():
+    assert erf2inv(0, 0) is S.Zero
+    assert erf2inv(0, 1) is S.Infinity
+    assert erf2inv(1, 0) is S.One
+    assert erf2inv(0, y) == erfinv(y)
+    assert erf2inv(oo, y) == erfcinv(-y)
+    assert erf2inv(x, 0) == x
+    assert erf2inv(x, oo) == erfinv(x)
+    assert erf2inv(nan, 0) is nan
+    assert erf2inv(0, nan) is nan
+
+    assert erf2inv(x, y).diff(x) == exp(-x**2 + erf2inv(x, y)**2)
+    assert erf2inv(x, y).diff(y) == sqrt(pi)*exp(erf2inv(x, y)**2)/2
+    raises(ArgumentIndexError, lambda: erf2inv(x, y).fdiff(3))
+
+
+# NOTE we multiply by exp_polar(I*pi) and need this to be on the principal
+# branch, hence take x in the lower half plane (d=0).
+
+
+def mytn(expr1, expr2, expr3, x, d=0):
+    from sympy.core.random import verify_numerically, random_complex_number
+    subs = {}
+    for a in expr1.free_symbols:
+        if a != x:
+            subs[a] = random_complex_number()
+    return expr2 == expr3 and verify_numerically(expr1.subs(subs),
+                                               expr2.subs(subs), x, d=d)
+
+
+def mytd(expr1, expr2, x):
+    from sympy.core.random import test_derivative_numerically, \
+        random_complex_number
+    subs = {}
+    for a in expr1.free_symbols:
+        if a != x:
+            subs[a] = random_complex_number()
+    return expr1.diff(x) == expr2 and test_derivative_numerically(expr1.subs(subs), x)
+
+
+def tn_branch(func, s=None):
+    from sympy.core.random import uniform
+
+    def fn(x):
+        if s is None:
+            return func(x)
+        return func(s, x)
+    c = uniform(1, 5)
+    expr = fn(c*exp_polar(I*pi)) - fn(c*exp_polar(-I*pi))
+    eps = 1e-15
+    expr2 = fn(-c + eps*I) - fn(-c - eps*I)
+    return abs(expr.n() - expr2.n()).n() < 1e-10
+
+
+def test_ei():
+    assert Ei(0) is S.NegativeInfinity
+    assert Ei(oo) is S.Infinity
+    assert Ei(-oo) is S.Zero
+
+    assert tn_branch(Ei)
+    assert mytd(Ei(x), exp(x)/x, x)
+    assert mytn(Ei(x), Ei(x).rewrite(uppergamma),
+                -uppergamma(0, x*polar_lift(-1)) - I*pi, x)
+    assert mytn(Ei(x), Ei(x).rewrite(expint),
+                -expint(1, x*polar_lift(-1)) - I*pi, x)
+    assert Ei(x).rewrite(expint).rewrite(Ei) == Ei(x)
+    assert Ei(x*exp_polar(2*I*pi)) == Ei(x) + 2*I*pi
+    assert Ei(x*exp_polar(-2*I*pi)) == Ei(x) - 2*I*pi
+
+    assert mytn(Ei(x), Ei(x).rewrite(Shi), Chi(x) + Shi(x), x)
+    assert mytn(Ei(x*polar_lift(I)), Ei(x*polar_lift(I)).rewrite(Si),
+                Ci(x) + I*Si(x) + I*pi/2, x)
+
+    assert Ei(log(x)).rewrite(li) == li(x)
+    assert Ei(2*log(x)).rewrite(li) == li(x**2)
+
+    assert gruntz(Ei(x+exp(-x))*exp(-x)*x, x, oo) == 1
+
+    assert Ei(x).series(x) == EulerGamma + log(x) + x + x**2/4 + \
+        x**3/18 + x**4/96 + x**5/600 + O(x**6)
+    assert Ei(x).series(x, 1, 3) == Ei(1) + E*(x - 1) + O((x - 1)**3, (x, 1))
+    assert Ei(x).series(x, oo) == \
+        (120/x**5 + 24/x**4 + 6/x**3 + 2/x**2 + 1/x + 1 + O(x**(-6), (x, oo)))*exp(x)/x
+
+    assert str(Ei(cos(2)).evalf(n=10)) == '-0.6760647401'
+    raises(ArgumentIndexError, lambda: Ei(x).fdiff(2))
+
+
+def test_expint():
+    assert mytn(expint(x, y), expint(x, y).rewrite(uppergamma),
+                y**(x - 1)*uppergamma(1 - x, y), x)
+    assert mytd(
+        expint(x, y), -y**(x - 1)*meijerg([], [1, 1], [0, 0, 1 - x], [], y), x)
+    assert mytd(expint(x, y), -expint(x - 1, y), y)
+    assert mytn(expint(1, x), expint(1, x).rewrite(Ei),
+                -Ei(x*polar_lift(-1)) + I*pi, x)
+
+    assert expint(-4, x) == exp(-x)/x + 4*exp(-x)/x**2 + 12*exp(-x)/x**3 \
+        + 24*exp(-x)/x**4 + 24*exp(-x)/x**5
+    assert expint(Rational(-3, 2), x) == \
+        exp(-x)/x + 3*exp(-x)/(2*x**2) + 3*sqrt(pi)*erfc(sqrt(x))/(4*x**S('5/2'))
+
+    assert tn_branch(expint, 1)
+    assert tn_branch(expint, 2)
+    assert tn_branch(expint, 3)
+    assert tn_branch(expint, 1.7)
+    assert tn_branch(expint, pi)
+
+    assert expint(y, x*exp_polar(2*I*pi)) == \
+        x**(y - 1)*(exp(2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x)
+    assert expint(y, x*exp_polar(-2*I*pi)) == \
+        x**(y - 1)*(exp(-2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x)
+    assert expint(2, x*exp_polar(2*I*pi)) == 2*I*pi*x + expint(2, x)
+    assert expint(2, x*exp_polar(-2*I*pi)) == -2*I*pi*x + expint(2, x)
+    assert expint(1, x).rewrite(Ei).rewrite(expint) == expint(1, x)
+    assert expint(x, y).rewrite(Ei) == expint(x, y)
+    assert expint(x, y).rewrite(Ci) == expint(x, y)
+
+    assert mytn(E1(x), E1(x).rewrite(Shi), Shi(x) - Chi(x), x)
+    assert mytn(E1(polar_lift(I)*x), E1(polar_lift(I)*x).rewrite(Si),
+                -Ci(x) + I*Si(x) - I*pi/2, x)
+
+    assert mytn(expint(2, x), expint(2, x).rewrite(Ei).rewrite(expint),
+                -x*E1(x) + exp(-x), x)
+    assert mytn(expint(3, x), expint(3, x).rewrite(Ei).rewrite(expint),
+                x**2*E1(x)/2 + (1 - x)*exp(-x)/2, x)
+
+    assert expint(Rational(3, 2), z).nseries(z) == \
+        2 + 2*z - z**2/3 + z**3/15 - z**4/84 + z**5/540 - \
+        2*sqrt(pi)*sqrt(z) + O(z**6)
+
+    assert E1(z).series(z) == -EulerGamma - log(z) + z - \
+        z**2/4 + z**3/18 - z**4/96 + z**5/600 + O(z**6)
+
+    assert expint(4, z).series(z) == Rational(1, 3) - z/2 + z**2/2 + \
+        z**3*(log(z)/6 - Rational(11, 36) + EulerGamma/6 - I*pi/6) - z**4/24 + \
+        z**5/240 + O(z**6)
+
+    assert expint(n, x).series(x, oo, n=3) == \
+        (n*(n + 1)/x**2 - n/x + 1 + O(x**(-3), (x, oo)))*exp(-x)/x
+
+    assert expint(z, y).series(z, 0, 2) == exp(-y)/y - z*meijerg(((), (1, 1)),
+                                  ((0, 0, 1), ()), y)/y + O(z**2)
+    raises(ArgumentIndexError, lambda: expint(x, y).fdiff(3))
+
+    neg = Symbol('neg', negative=True)
+    assert Ei(neg).rewrite(Si) == Shi(neg) + Chi(neg) - I*pi
+
+
+def test__eis():
+    assert _eis(z).diff(z) == -_eis(z) + 1/z
+
+    assert _eis(1/z).series(z) == \
+        z + z**2 + 2*z**3 + 6*z**4 + 24*z**5 + O(z**6)
+
+    assert Ei(z).rewrite('tractable') == exp(z)*_eis(z)
+    assert li(z).rewrite('tractable') == z*_eis(log(z))
+
+    assert _eis(z).rewrite('intractable') == exp(-z)*Ei(z)
+
+    assert expand(li(z).rewrite('tractable').diff(z).rewrite('intractable')) \
+        == li(z).diff(z)
+
+    assert expand(Ei(z).rewrite('tractable').diff(z).rewrite('intractable')) \
+        == Ei(z).diff(z)
+
+    assert _eis(z).series(z, n=3) == EulerGamma + log(z) + z*(-log(z) - \
+        EulerGamma + 1) + z**2*(log(z)/2 - Rational(3, 4) + EulerGamma/2)\
+        + O(z**3*log(z))
+    raises(ArgumentIndexError, lambda: _eis(z).fdiff(2))
+
+
+def tn_arg(func):
+    def test(arg, e1, e2):
+        from sympy.core.random import uniform
+        v = uniform(1, 5)
+        v1 = func(arg*x).subs(x, v).n()
+        v2 = func(e1*v + e2*1e-15).n()
+        return abs(v1 - v2).n() < 1e-10
+    return test(exp_polar(I*pi/2), I, 1) and \
+        test(exp_polar(-I*pi/2), -I, 1) and \
+        test(exp_polar(I*pi), -1, I) and \
+        test(exp_polar(-I*pi), -1, -I)
+
+
+def test_li():
+    z = Symbol("z")
+    zr = Symbol("z", real=True)
+    zp = Symbol("z", positive=True)
+    zn = Symbol("z", negative=True)
+
+    assert li(0) is S.Zero
+    assert li(1) is -oo
+    assert li(oo) is oo
+
+    assert isinstance(li(z), li)
+    assert unchanged(li, -zp)
+    assert unchanged(li, zn)
+
+    assert diff(li(z), z) == 1/log(z)
+
+    assert conjugate(li(z)) == li(conjugate(z))
+    assert conjugate(li(-zr)) == li(-zr)
+    assert unchanged(conjugate, li(-zp))
+    assert unchanged(conjugate, li(zn))
+
+    assert li(z).rewrite(Li) == Li(z) + li(2)
+    assert li(z).rewrite(Ei) == Ei(log(z))
+    assert li(z).rewrite(uppergamma) == (-log(1/log(z))/2 - log(-log(z)) +
+                                         log(log(z))/2 - expint(1, -log(z)))
+    assert li(z).rewrite(Si) == (-log(I*log(z)) - log(1/log(z))/2 +
+                                 log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
+    assert li(z).rewrite(Ci) == (-log(I*log(z)) - log(1/log(z))/2 +
+                                 log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
+    assert li(z).rewrite(Shi) == (-log(1/log(z))/2 + log(log(z))/2 +
+                                  Chi(log(z)) - Shi(log(z)))
+    assert li(z).rewrite(Chi) == (-log(1/log(z))/2 + log(log(z))/2 +
+                                  Chi(log(z)) - Shi(log(z)))
+    assert li(z).rewrite(hyper) ==(log(z)*hyper((1, 1), (2, 2), log(z)) -
+                                   log(1/log(z))/2 + log(log(z))/2 + EulerGamma)
+    assert li(z).rewrite(meijerg) == (-log(1/log(z))/2 - log(-log(z)) + log(log(z))/2 -
+                                      meijerg(((), (1,)), ((0, 0), ()), -log(z)))
+
+    assert gruntz(1/li(z), z, oo) is S.Zero
+    assert li(z).series(z) == log(z)**5/600 + log(z)**4/96 + log(z)**3/18 + log(z)**2/4 + \
+            log(z) + log(log(z)) + EulerGamma
+    raises(ArgumentIndexError, lambda: li(z).fdiff(2))
+
+
+def test_Li():
+    assert Li(2) is S.Zero
+    assert Li(oo) is oo
+
+    assert isinstance(Li(z), Li)
+
+    assert diff(Li(z), z) == 1/log(z)
+
+    assert gruntz(1/Li(z), z, oo) is S.Zero
+    assert Li(z).rewrite(li) == li(z) - li(2)
+    assert Li(z).series(z) == \
+        log(z)**5/600 + log(z)**4/96 + log(z)**3/18 + log(z)**2/4 + log(z) + log(log(z)) - li(2) + EulerGamma
+    raises(ArgumentIndexError, lambda: Li(z).fdiff(2))
+
+
+def test_si():
+    assert Si(I*x) == I*Shi(x)
+    assert Shi(I*x) == I*Si(x)
+    assert Si(-I*x) == -I*Shi(x)
+    assert Shi(-I*x) == -I*Si(x)
+    assert Si(-x) == -Si(x)
+    assert Shi(-x) == -Shi(x)
+    assert Si(exp_polar(2*pi*I)*x) == Si(x)
+    assert Si(exp_polar(-2*pi*I)*x) == Si(x)
+    assert Shi(exp_polar(2*pi*I)*x) == Shi(x)
+    assert Shi(exp_polar(-2*pi*I)*x) == Shi(x)
+
+    assert Si(oo) == pi/2
+    assert Si(-oo) == -pi/2
+    assert Shi(oo) is oo
+    assert Shi(-oo) is -oo
+
+    assert mytd(Si(x), sin(x)/x, x)
+    assert mytd(Shi(x), sinh(x)/x, x)
+
+    assert mytn(Si(x), Si(x).rewrite(Ei),
+                -I*(-Ei(x*exp_polar(-I*pi/2))/2
+               + Ei(x*exp_polar(I*pi/2))/2 - I*pi) + pi/2, x)
+    assert mytn(Si(x), Si(x).rewrite(expint),
+                -I*(-expint(1, x*exp_polar(-I*pi/2))/2 +
+                    expint(1, x*exp_polar(I*pi/2))/2) + pi/2, x)
+    assert mytn(Shi(x), Shi(x).rewrite(Ei),
+                Ei(x)/2 - Ei(x*exp_polar(I*pi))/2 + I*pi/2, x)
+    assert mytn(Shi(x), Shi(x).rewrite(expint),
+                expint(1, x)/2 - expint(1, x*exp_polar(I*pi))/2 - I*pi/2, x)
+
+    assert tn_arg(Si)
+    assert tn_arg(Shi)
+
+    assert Si(x)._eval_as_leading_term(x) == x
+    assert Si(2*x)._eval_as_leading_term(x) == 2*x
+    assert Si(sin(x))._eval_as_leading_term(x) == x
+    assert Si(x + 1)._eval_as_leading_term(x) == Si(1)
+    assert Si(1/x)._eval_as_leading_term(x, cdir=1) == \
+        Si(1/x)._eval_as_leading_term(x, cdir=-1) == Si(1/x)
+
+    assert Si(x).nseries(x, n=8) == \
+        x - x**3/18 + x**5/600 - x**7/35280 + O(x**8)
+    assert Shi(x).nseries(x, n=8) == \
+        x + x**3/18 + x**5/600 + x**7/35280 + O(x**8)
+    assert Si(sin(x)).nseries(x, n=5) == x - 2*x**3/9 + O(x**5)
+    assert Si(x).nseries(x, 1, n=3) == \
+        Si(1) + (x - 1)*sin(1) + (x - 1)**2*(-sin(1)/2 + cos(1)/2) + O((x - 1)**3, (x, 1))
+
+    assert Si(x).series(x, oo) == -sin(x)*(-6/x**4 + x**(-2) + O(x**(-6), (x, oo))) - \
+        cos(x)*(24/x**5 - 2/x**3 + 1/x + O(x**(-6), (x, oo))) + pi/2
+
+    t = Symbol('t', Dummy=True)
+    assert Si(x).rewrite(sinc).dummy_eq(Integral(sinc(t), (t, 0, x)))
+
+    assert limit(Shi(x), x, S.Infinity) == S.Infinity
+    assert limit(Shi(x), x, S.NegativeInfinity) == S.NegativeInfinity
+
+
+def test_ci():
+    m1 = exp_polar(I*pi)
+    m1_ = exp_polar(-I*pi)
+    pI = exp_polar(I*pi/2)
+    mI = exp_polar(-I*pi/2)
+
+    assert Ci(m1*x) == Ci(x) + I*pi
+    assert Ci(m1_*x) == Ci(x) - I*pi
+    assert Ci(pI*x) == Chi(x) + I*pi/2
+    assert Ci(mI*x) == Chi(x) - I*pi/2
+    assert Chi(m1*x) == Chi(x) + I*pi
+    assert Chi(m1_*x) == Chi(x) - I*pi
+    assert Chi(pI*x) == Ci(x) + I*pi/2
+    assert Chi(mI*x) == Ci(x) - I*pi/2
+    assert Ci(exp_polar(2*I*pi)*x) == Ci(x) + 2*I*pi
+    assert Chi(exp_polar(-2*I*pi)*x) == Chi(x) - 2*I*pi
+    assert Chi(exp_polar(2*I*pi)*x) == Chi(x) + 2*I*pi
+    assert Ci(exp_polar(-2*I*pi)*x) == Ci(x) - 2*I*pi
+
+    assert Ci(oo) is S.Zero
+    assert Ci(-oo) == I*pi
+    assert Chi(oo) is oo
+    assert Chi(-oo) is oo
+
+    assert mytd(Ci(x), cos(x)/x, x)
+    assert mytd(Chi(x), cosh(x)/x, x)
+
+    assert mytn(Ci(x), Ci(x).rewrite(Ei),
+                Ei(x*exp_polar(-I*pi/2))/2 + Ei(x*exp_polar(I*pi/2))/2, x)
+    assert mytn(Chi(x), Chi(x).rewrite(Ei),
+                Ei(x)/2 + Ei(x*exp_polar(I*pi))/2 - I*pi/2, x)
+
+    assert tn_arg(Ci)
+    assert tn_arg(Chi)
+
+    assert Ci(x).nseries(x, n=4) == \
+        EulerGamma + log(x) - x**2/4 + O(x**4)
+    assert Chi(x).nseries(x, n=4) == \
+        EulerGamma + log(x) + x**2/4 + O(x**4)
+
+    assert Ci(x).series(x, oo) == -cos(x)*(-6/x**4 + x**(-2) + O(x**(-6), (x, oo))) + \
+        sin(x)*(24/x**5 - 2/x**3 + 1/x + O(x**(-6), (x, oo)))
+
+    assert Ci(x).series(x, -oo) == -cos(x)*(-6/x**4 + x**(-2) + O(x**(-6), (x, -oo))) + \
+        sin(x)*(24/x**5 - 2/x**3 + 1/x + O(x**(-6), (x, -oo))) + I*pi
+
+    assert limit(log(x) - Ci(2*x), x, 0) == -log(2) - EulerGamma
+    assert Ci(x).rewrite(uppergamma) == -expint(1, x*exp_polar(-I*pi/2))/2 -\
+                                        expint(1, x*exp_polar(I*pi/2))/2
+    assert Ci(x).rewrite(expint) == -expint(1, x*exp_polar(-I*pi/2))/2 -\
+                                        expint(1, x*exp_polar(I*pi/2))/2
+    raises(ArgumentIndexError, lambda: Ci(x).fdiff(2))
+
+
+def test_fresnel():
+    assert fresnels(0) is S.Zero
+    assert fresnels(oo) is S.Half
+    assert fresnels(-oo) == Rational(-1, 2)
+    assert fresnels(I*oo) == -I*S.Half
+
+    assert unchanged(fresnels, z)
+    assert fresnels(-z) == -fresnels(z)
+    assert fresnels(I*z) == -I*fresnels(z)
+    assert fresnels(-I*z) == I*fresnels(z)
+
+    assert conjugate(fresnels(z)) == fresnels(conjugate(z))
+
+    assert fresnels(z).diff(z) == sin(pi*z**2/2)
+
+    assert fresnels(z).rewrite(erf) == (S.One + I)/4 * (
+        erf((S.One + I)/2*sqrt(pi)*z) - I*erf((S.One - I)/2*sqrt(pi)*z))
+
+    assert fresnels(z).rewrite(hyper) == \
+        pi*z**3/6 * hyper([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)], -pi**2*z**4/16)
+
+    assert fresnels(z).series(z, n=15) == \
+        pi*z**3/6 - pi**3*z**7/336 + pi**5*z**11/42240 + O(z**15)
+
+    assert fresnels(w).is_extended_real is True
+    assert fresnels(w).is_finite is True
+
+    assert fresnels(z).is_extended_real is None
+    assert fresnels(z).is_finite is None
+
+    assert fresnels(z).as_real_imag() == (fresnels(re(z) - I*im(z))/2 +
+                    fresnels(re(z) + I*im(z))/2,
+                    -I*(-fresnels(re(z) - I*im(z)) + fresnels(re(z) + I*im(z)))/2)
+
+    assert fresnels(z).as_real_imag(deep=False) == (fresnels(re(z) - I*im(z))/2 +
+                    fresnels(re(z) + I*im(z))/2,
+                    -I*(-fresnels(re(z) - I*im(z)) + fresnels(re(z) + I*im(z)))/2)
+
+    assert fresnels(w).as_real_imag() == (fresnels(w), 0)
+    assert fresnels(w).as_real_imag(deep=True) == (fresnels(w), 0)
+
+    assert fresnels(2 + 3*I).as_real_imag() == (
+            fresnels(2 + 3*I)/2 + fresnels(2 - 3*I)/2,
+            -I*(fresnels(2 + 3*I) - fresnels(2 - 3*I))/2
+            )
+
+    assert expand_func(integrate(fresnels(z), z)) == \
+        z*fresnels(z) + cos(pi*z**2/2)/pi
+
+    assert fresnels(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(9, 4) * \
+        meijerg(((), (1,)), ((Rational(3, 4),),
+        (Rational(1, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(3, 4)*(z**2)**Rational(3, 4))
+
+    assert fresnelc(0) is S.Zero
+    assert fresnelc(oo) == S.Half
+    assert fresnelc(-oo) == Rational(-1, 2)
+    assert fresnelc(I*oo) == I*S.Half
+
+    assert unchanged(fresnelc, z)
+    assert fresnelc(-z) == -fresnelc(z)
+    assert fresnelc(I*z) == I*fresnelc(z)
+    assert fresnelc(-I*z) == -I*fresnelc(z)
+
+    assert conjugate(fresnelc(z)) == fresnelc(conjugate(z))
+
+    assert fresnelc(z).diff(z) == cos(pi*z**2/2)
+
+    assert fresnelc(z).rewrite(erf) == (S.One - I)/4 * (
+        erf((S.One + I)/2*sqrt(pi)*z) + I*erf((S.One - I)/2*sqrt(pi)*z))
+
+    assert fresnelc(z).rewrite(hyper) == \
+        z * hyper([Rational(1, 4)], [S.Half, Rational(5, 4)], -pi**2*z**4/16)
+
+    assert fresnelc(w).is_extended_real is True
+
+    assert fresnelc(z).as_real_imag() == \
+        (fresnelc(re(z) - I*im(z))/2 + fresnelc(re(z) + I*im(z))/2,
+         -I*(-fresnelc(re(z) - I*im(z)) + fresnelc(re(z) + I*im(z)))/2)
+
+    assert fresnelc(z).as_real_imag(deep=False) == \
+        (fresnelc(re(z) - I*im(z))/2 + fresnelc(re(z) + I*im(z))/2,
+         -I*(-fresnelc(re(z) - I*im(z)) + fresnelc(re(z) + I*im(z)))/2)
+
+    assert fresnelc(2 + 3*I).as_real_imag() == (
+        fresnelc(2 - 3*I)/2 + fresnelc(2 + 3*I)/2,
+         -I*(fresnelc(2 + 3*I) - fresnelc(2 - 3*I))/2
+    )
+
+    assert expand_func(integrate(fresnelc(z), z)) == \
+        z*fresnelc(z) - sin(pi*z**2/2)/pi
+
+    assert fresnelc(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(3, 4) * \
+        meijerg(((), (1,)), ((Rational(1, 4),),
+        (Rational(3, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(1, 4)*(z**2)**Rational(1, 4))
+
+    from sympy.core.random import verify_numerically
+
+    verify_numerically(re(fresnels(z)), fresnels(z).as_real_imag()[0], z)
+    verify_numerically(im(fresnels(z)), fresnels(z).as_real_imag()[1], z)
+    verify_numerically(fresnels(z), fresnels(z).rewrite(hyper), z)
+    verify_numerically(fresnels(z), fresnels(z).rewrite(meijerg), z)
+
+    verify_numerically(re(fresnelc(z)), fresnelc(z).as_real_imag()[0], z)
+    verify_numerically(im(fresnelc(z)), fresnelc(z).as_real_imag()[1], z)
+    verify_numerically(fresnelc(z), fresnelc(z).rewrite(hyper), z)
+    verify_numerically(fresnelc(z), fresnelc(z).rewrite(meijerg), z)
+
+    raises(ArgumentIndexError, lambda: fresnels(z).fdiff(2))
+    raises(ArgumentIndexError, lambda: fresnelc(z).fdiff(2))
+
+    assert fresnels(x).taylor_term(-1, x) is S.Zero
+    assert fresnelc(x).taylor_term(-1, x) is S.Zero
+    assert fresnelc(x).taylor_term(1, x) == -pi**2*x**5/40
+
+
+def test_fresnel_series():
+    assert fresnelc(z).series(z, n=15) == \
+        z - pi**2*z**5/40 + pi**4*z**9/3456 - pi**6*z**13/599040 + O(z**15)
+
+    # issues 6510, 10102
+    fs = (S.Half - sin(pi*z**2/2)/(pi**2*z**3)
+        + (-1/(pi*z) + 3/(pi**3*z**5))*cos(pi*z**2/2))
+    fc = (S.Half - cos(pi*z**2/2)/(pi**2*z**3)
+        + (1/(pi*z) - 3/(pi**3*z**5))*sin(pi*z**2/2))
+    assert fresnels(z).series(z, oo) == fs + O(z**(-6), (z, oo))
+    assert fresnelc(z).series(z, oo) == fc + O(z**(-6), (z, oo))
+    assert (fresnels(z).series(z, -oo) + fs.subs(z, -z)).expand().is_Order
+    assert (fresnelc(z).series(z, -oo) + fc.subs(z, -z)).expand().is_Order
+    assert (fresnels(1/z).series(z) - fs.subs(z, 1/z)).expand().is_Order
+    assert (fresnelc(1/z).series(z) - fc.subs(z, 1/z)).expand().is_Order
+    assert ((2*fresnels(3*z)).series(z, oo) - 2*fs.subs(z, 3*z)).expand().is_Order
+    assert ((3*fresnelc(2*z)).series(z, oo) - 3*fc.subs(z, 2*z)).expand().is_Order
+
+
+def test_integral_rewrites(): #issues 26134, 26144, 26306
+    assert expint(n, x).rewrite(Integral).dummy_eq(Integral(t**-n * exp(-t*x), (t, 1, oo)))
+    assert Si(x).rewrite(Integral).dummy_eq(Integral(sinc(t), (t, 0, x)))
+    assert Ci(x).rewrite(Integral).dummy_eq(log(x) - Integral((1 - cos(t))/t, (t, 0, x)) + EulerGamma)
+    assert fresnels(x).rewrite(Integral).dummy_eq(Integral(sin(pi*t**2/2), (t, 0, x)))
+    assert fresnelc(x).rewrite(Integral).dummy_eq(Integral(cos(pi*t**2/2), (t, 0, x)))
+    assert Ei(x).rewrite(Integral).dummy_eq(Integral(exp(t)/t, (t, -oo, x)))
+    assert fresnels(x).diff(x) == fresnels(x).rewrite(Integral).diff(x)
+    assert fresnelc(x).diff(x) == fresnelc(x).rewrite(Integral).diff(x)

rl/Lib/site-packages/sympy/functions/special/tests/test_gamma_functions.py

@@ -0,0 +1,741 @@
+from sympy.core.function import expand_func, Subs
+from sympy.core import EulerGamma
+from sympy.core.numbers import (I, Rational, nan, oo, pi, zoo)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol)
+from sympy.functions.combinatorial.factorials import factorial
+from sympy.functions.combinatorial.numbers import harmonic
+from sympy.functions.elementary.complexes import (Abs, conjugate, im, re)
+from sympy.functions.elementary.exponential import (exp, exp_polar, log)
+from sympy.functions.elementary.hyperbolic import tanh
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin, atan)
+from sympy.functions.special.error_functions import (Ei, erf, erfc)
+from sympy.functions.special.gamma_functions import (digamma, gamma, loggamma, lowergamma, multigamma, polygamma, trigamma, uppergamma)
+from sympy.functions.special.zeta_functions import zeta
+from sympy.series.order import O
+
+from sympy.core.expr import unchanged
+from sympy.core.function import ArgumentIndexError
+from sympy.testing.pytest import raises
+from sympy.core.random import (test_derivative_numerically as td,
+                                      random_complex_number as randcplx,
+                                      verify_numerically as tn)
+
+x = Symbol('x')
+y = Symbol('y')
+n = Symbol('n', integer=True)
+w = Symbol('w', real=True)
+
+def test_gamma():
+    assert gamma(nan) is nan
+    assert gamma(oo) is oo
+
+    assert gamma(-100) is zoo
+    assert gamma(0) is zoo
+    assert gamma(-100.0) is zoo
+
+    assert gamma(1) == 1
+    assert gamma(2) == 1
+    assert gamma(3) == 2
+
+    assert gamma(102) == factorial(101)
+
+    assert gamma(S.Half) == sqrt(pi)
+
+    assert gamma(Rational(3, 2)) == sqrt(pi)*S.Half
+    assert gamma(Rational(5, 2)) == sqrt(pi)*Rational(3, 4)
+    assert gamma(Rational(7, 2)) == sqrt(pi)*Rational(15, 8)
+
+    assert gamma(Rational(-1, 2)) == -2*sqrt(pi)
+    assert gamma(Rational(-3, 2)) == sqrt(pi)*Rational(4, 3)
+    assert gamma(Rational(-5, 2)) == sqrt(pi)*Rational(-8, 15)
+
+    assert gamma(Rational(-15, 2)) == sqrt(pi)*Rational(256, 2027025)
+
+    assert gamma(Rational(
+        -11, 8)).expand(func=True) == Rational(64, 33)*gamma(Rational(5, 8))
+    assert gamma(Rational(
+        -10, 3)).expand(func=True) == Rational(81, 280)*gamma(Rational(2, 3))
+    assert gamma(Rational(
+        14, 3)).expand(func=True) == Rational(880, 81)*gamma(Rational(2, 3))
+    assert gamma(Rational(
+        17, 7)).expand(func=True) == Rational(30, 49)*gamma(Rational(3, 7))
+    assert gamma(Rational(
+        19, 8)).expand(func=True) == Rational(33, 64)*gamma(Rational(3, 8))
+
+    assert gamma(x).diff(x) == gamma(x)*polygamma(0, x)
+
+    assert gamma(x - 1).expand(func=True) == gamma(x)/(x - 1)
+    assert gamma(x + 2).expand(func=True, mul=False) == x*(x + 1)*gamma(x)
+
+    assert conjugate(gamma(x)) == gamma(conjugate(x))
+
+    assert expand_func(gamma(x + Rational(3, 2))) == \
+        (x + S.Half)*gamma(x + S.Half)
+
+    assert expand_func(gamma(x - S.Half)) == \
+        gamma(S.Half + x)/(x - S.Half)
+
+    # Test a bug:
+    assert expand_func(gamma(x + Rational(3, 4))) == gamma(x + Rational(3, 4))
+
+    # XXX: Not sure about these tests. I can fix them by defining e.g.
+    # exp_polar.is_integer but I'm not sure if that makes sense.
+    assert gamma(3*exp_polar(I*pi)/4).is_nonnegative is False
+    assert gamma(3*exp_polar(I*pi)/4).is_extended_nonpositive is True
+
+    y = Symbol('y', nonpositive=True, integer=True)
+    assert gamma(y).is_real == False
+    y = Symbol('y', positive=True, noninteger=True)
+    assert gamma(y).is_real == True
+
+    assert gamma(-1.0, evaluate=False).is_real == False
+    assert gamma(0, evaluate=False).is_real == False
+    assert gamma(-2, evaluate=False).is_real == False
+
+
+def test_gamma_rewrite():
+    assert gamma(n).rewrite(factorial) == factorial(n - 1)
+
+
+def test_gamma_series():
+    assert gamma(x + 1).series(x, 0, 3) == \
+        1 - EulerGamma*x + x**2*(EulerGamma**2/2 + pi**2/12) + O(x**3)
+    assert gamma(x).series(x, -1, 3) == \
+        -1/(x + 1) + EulerGamma - 1 + (x + 1)*(-1 - pi**2/12 - EulerGamma**2/2 + \
+       EulerGamma) + (x + 1)**2*(-1 - pi**2/12 - EulerGamma**2/2 + EulerGamma**3/6 - \
+       polygamma(2, 1)/6 + EulerGamma*pi**2/12 + EulerGamma) + O((x + 1)**3, (x, -1))
+
+
+def tn_branch(s, func):
+    from sympy.core.random import uniform
+    c = uniform(1, 5)
+    expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
+    eps = 1e-15
+    expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
+    return abs(expr.n() - expr2.n()).n() < 1e-10
+
+
+def test_lowergamma():
+    from sympy.functions.special.error_functions import expint
+    from sympy.functions.special.hyper import meijerg
+    assert lowergamma(x, 0) == 0
+    assert lowergamma(x, y).diff(y) == y**(x - 1)*exp(-y)
+    assert td(lowergamma(randcplx(), y), y)
+    assert td(lowergamma(x, randcplx()), x)
+    assert lowergamma(x, y).diff(x) == \
+        gamma(x)*digamma(x) - uppergamma(x, y)*log(y) \
+        - meijerg([], [1, 1], [0, 0, x], [], y)
+
+    assert lowergamma(S.Half, x) == sqrt(pi)*erf(sqrt(x))
+    assert not lowergamma(S.Half - 3, x).has(lowergamma)
+    assert not lowergamma(S.Half + 3, x).has(lowergamma)
+    assert lowergamma(S.Half, x, evaluate=False).has(lowergamma)
+    assert tn(lowergamma(S.Half + 3, x, evaluate=False),
+              lowergamma(S.Half + 3, x), x)
+    assert tn(lowergamma(S.Half - 3, x, evaluate=False),
+              lowergamma(S.Half - 3, x), x)
+
+    assert tn_branch(-3, lowergamma)
+    assert tn_branch(-4, lowergamma)
+    assert tn_branch(Rational(1, 3), lowergamma)
+    assert tn_branch(pi, lowergamma)
+    assert lowergamma(3, exp_polar(4*pi*I)*x) == lowergamma(3, x)
+    assert lowergamma(y, exp_polar(5*pi*I)*x) == \
+        exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I))
+    assert lowergamma(-2, exp_polar(5*pi*I)*x) == \
+        lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I
+
+    assert conjugate(lowergamma(x, y)) == lowergamma(conjugate(x), conjugate(y))
+    assert conjugate(lowergamma(x, 0)) == 0
+    assert unchanged(conjugate, lowergamma(x, -oo))
+
+    assert lowergamma(0, x)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(S(1)/3, x)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(1, x, evaluate=False)._eval_is_meromorphic(x, 0) == True
+    assert lowergamma(x, x)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(x + 1, x)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(1/x, x)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(0, x + 1)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(S(1)/3, x + 1)._eval_is_meromorphic(x, 0) == True
+    assert lowergamma(1, x + 1, evaluate=False)._eval_is_meromorphic(x, 0) == True
+    assert lowergamma(x, x + 1)._eval_is_meromorphic(x, 0) == True
+    assert lowergamma(x + 1, x + 1)._eval_is_meromorphic(x, 0) == True
+    assert lowergamma(1/x, x + 1)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(0, 1/x)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(S(1)/3, 1/x)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(1, 1/x, evaluate=False)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(x, 1/x)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(x + 1, 1/x)._eval_is_meromorphic(x, 0) == False
+    assert lowergamma(1/x, 1/x)._eval_is_meromorphic(x, 0) == False
+
+    assert lowergamma(x, 2).series(x, oo, 3) == \
+        2**x*(1 + 2/(x + 1))*exp(-2)/x + O(exp(x*log(2))/x**3, (x, oo))
+
+    assert lowergamma(
+        x, y).rewrite(expint) == -y**x*expint(-x + 1, y) + gamma(x)
+    k = Symbol('k', integer=True)
+    assert lowergamma(
+        k, y).rewrite(expint) == -y**k*expint(-k + 1, y) + gamma(k)
+    k = Symbol('k', integer=True, positive=False)
+    assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)
+    assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)
+
+    assert lowergamma(70, 6) == factorial(69) - 69035724522603011058660187038367026272747334489677105069435923032634389419656200387949342530805432320 * exp(-6)
+    assert (lowergamma(S(77) / 2, 6) - lowergamma(S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
+    assert (lowergamma(-S(77) / 2, 6) - lowergamma(-S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
+
+
+def test_uppergamma():
+    from sympy.functions.special.error_functions import expint
+    from sympy.functions.special.hyper import meijerg
+    assert uppergamma(4, 0) == 6
+    assert uppergamma(x, y).diff(y) == -y**(x - 1)*exp(-y)
+    assert td(uppergamma(randcplx(), y), y)
+    assert uppergamma(x, y).diff(x) == \
+        uppergamma(x, y)*log(y) + meijerg([], [1, 1], [0, 0, x], [], y)
+    assert td(uppergamma(x, randcplx()), x)
+
+    p = Symbol('p', positive=True)
+    assert uppergamma(0, p) == -Ei(-p)
+    assert uppergamma(p, 0) == gamma(p)
+    assert uppergamma(S.Half, x) == sqrt(pi)*erfc(sqrt(x))
+    assert not uppergamma(S.Half - 3, x).has(uppergamma)
+    assert not uppergamma(S.Half + 3, x).has(uppergamma)
+    assert uppergamma(S.Half, x, evaluate=False).has(uppergamma)
+    assert tn(uppergamma(S.Half + 3, x, evaluate=False),
+              uppergamma(S.Half + 3, x), x)
+    assert tn(uppergamma(S.Half - 3, x, evaluate=False),
+              uppergamma(S.Half - 3, x), x)
+
+    assert unchanged(uppergamma, x, -oo)
+    assert unchanged(uppergamma, x, 0)
+
+    assert tn_branch(-3, uppergamma)
+    assert tn_branch(-4, uppergamma)
+    assert tn_branch(Rational(1, 3), uppergamma)
+    assert tn_branch(pi, uppergamma)
+    assert uppergamma(3, exp_polar(4*pi*I)*x) == uppergamma(3, x)
+    assert uppergamma(y, exp_polar(5*pi*I)*x) == \
+        exp(4*I*pi*y)*uppergamma(y, x*exp_polar(pi*I)) + \
+        gamma(y)*(1 - exp(4*pi*I*y))
+    assert uppergamma(-2, exp_polar(5*pi*I)*x) == \
+        uppergamma(-2, x*exp_polar(I*pi)) - 2*pi*I
+
+    assert uppergamma(-2, x) == expint(3, x)/x**2
+
+    assert conjugate(uppergamma(x, y)) == uppergamma(conjugate(x), conjugate(y))
+    assert unchanged(conjugate, uppergamma(x, -oo))
+
+    assert uppergamma(x, y).rewrite(expint) == y**x*expint(-x + 1, y)
+    assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y)
+
+    assert uppergamma(70, 6) == 69035724522603011058660187038367026272747334489677105069435923032634389419656200387949342530805432320*exp(-6)
+    assert (uppergamma(S(77) / 2, 6) - uppergamma(S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
+    assert (uppergamma(-S(77) / 2, 6) - uppergamma(-S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
+
+
+def test_polygamma():
+    assert polygamma(n, nan) is nan
+
+    assert polygamma(0, oo) is oo
+    assert polygamma(0, -oo) is oo
+    assert polygamma(0, I*oo) is oo
+    assert polygamma(0, -I*oo) is oo
+    assert polygamma(1, oo) == 0
+    assert polygamma(5, oo) == 0
+
+    assert polygamma(0, -9) is zoo
+
+    assert polygamma(0, -9) is zoo
+    assert polygamma(0, -1) is zoo
+    assert polygamma(Rational(3, 2), -1) is zoo
+
+    assert polygamma(0, 0) is zoo
+
+    assert polygamma(0, 1) == -EulerGamma
+    assert polygamma(0, 7) == Rational(49, 20) - EulerGamma
+
+    assert polygamma(1, 1) == pi**2/6
+    assert polygamma(1, 2) == pi**2/6 - 1
+    assert polygamma(1, 3) == pi**2/6 - Rational(5, 4)
+    assert polygamma(3, 1) == pi**4 / 15
+    assert polygamma(3, 5) == 6*(Rational(-22369, 20736) + pi**4/90)
+    assert polygamma(5, 1) == 8 * pi**6 / 63
+
+    assert polygamma(1, S.Half) == pi**2 / 2
+    assert polygamma(2, S.Half) == -14*zeta(3)
+    assert polygamma(11, S.Half) == 176896*pi**12
+
+    def t(m, n):
+        x = S(m)/n
+        r = polygamma(0, x)
+        if r.has(polygamma):
+            return False
+        return abs(polygamma(0, x.n()).n() - r.n()).n() < 1e-10
+    assert t(1, 2)
+    assert t(3, 2)
+    assert t(-1, 2)
+    assert t(1, 4)
+    assert t(-3, 4)
+    assert t(1, 3)
+    assert t(4, 3)
+    assert t(3, 4)
+    assert t(2, 3)
+    assert t(123, 5)
+
+    assert polygamma(0, x).rewrite(zeta) == polygamma(0, x)
+    assert polygamma(1, x).rewrite(zeta) == zeta(2, x)
+    assert polygamma(2, x).rewrite(zeta) == -2*zeta(3, x)
+    assert polygamma(I, 2).rewrite(zeta) == polygamma(I, 2)
+    n1 = Symbol('n1')
+    n2 = Symbol('n2', real=True)
+    n3 = Symbol('n3', integer=True)
+    n4 = Symbol('n4', positive=True)
+    n5 = Symbol('n5', positive=True, integer=True)
+    assert polygamma(n1, x).rewrite(zeta) == polygamma(n1, x)
+    assert polygamma(n2, x).rewrite(zeta) == polygamma(n2, x)
+    assert polygamma(n3, x).rewrite(zeta) == polygamma(n3, x)
+    assert polygamma(n4, x).rewrite(zeta) == polygamma(n4, x)
+    assert polygamma(n5, x).rewrite(zeta) == (-1)**(n5 + 1) * factorial(n5) * zeta(n5 + 1, x)
+
+    assert polygamma(3, 7*x).diff(x) == 7*polygamma(4, 7*x)
+
+    assert polygamma(0, x).rewrite(harmonic) == harmonic(x - 1) - EulerGamma
+    assert polygamma(2, x).rewrite(harmonic) == 2*harmonic(x - 1, 3) - 2*zeta(3)
+    ni = Symbol("n", integer=True)
+    assert polygamma(ni, x).rewrite(harmonic) == (-1)**(ni + 1)*(-harmonic(x - 1, ni + 1)
+                                                                 + zeta(ni + 1))*factorial(ni)
+
+    # Polygamma of non-negative integer order is unbranched:
+    k = Symbol('n', integer=True, nonnegative=True)
+    assert polygamma(k, exp_polar(2*I*pi)*x) == polygamma(k, x)
+
+    # but negative integers are branched!
+    k = Symbol('n', integer=True)
+    assert polygamma(k, exp_polar(2*I*pi)*x).args == (k, exp_polar(2*I*pi)*x)
+
+    # Polygamma of order -1 is loggamma:
+    assert polygamma(-1, x) == loggamma(x) - log(2*pi) / 2
+
+    # But smaller orders are iterated integrals and don't have a special name
+    assert polygamma(-2, x).func is polygamma
+
+    # Test a bug
+    assert polygamma(0, -x).expand(func=True) == polygamma(0, -x)
+
+    assert polygamma(2, 2.5).is_positive == False
+    assert polygamma(2, -2.5).is_positive == False
+    assert polygamma(3, 2.5).is_positive == True
+    assert polygamma(3, -2.5).is_positive is True
+    assert polygamma(-2, -2.5).is_positive is None
+    assert polygamma(-3, -2.5).is_positive is None
+
+    assert polygamma(2, 2.5).is_negative == True
+    assert polygamma(3, 2.5).is_negative == False
+    assert polygamma(3, -2.5).is_negative == False
+    assert polygamma(2, -2.5).is_negative is True
+    assert polygamma(-2, -2.5).is_negative is None
+    assert polygamma(-3, -2.5).is_negative is None
+
+    assert polygamma(I, 2).is_positive is None
+    assert polygamma(I, 3).is_negative is None
+
+    # issue 17350
+    assert (I*polygamma(I, pi)).as_real_imag() == \
+           (-im(polygamma(I, pi)), re(polygamma(I, pi)))
+    assert (tanh(polygamma(I, 1))).rewrite(exp) == \
+           (exp(polygamma(I, 1)) - exp(-polygamma(I, 1)))/(exp(polygamma(I, 1)) + exp(-polygamma(I, 1)))
+    assert (I / polygamma(I, 4)).rewrite(exp) == \
+           I*exp(-I*atan(im(polygamma(I, 4))/re(polygamma(I, 4))))/Abs(polygamma(I, 4))
+
+    # issue 12569
+    assert unchanged(im, polygamma(0, I))
+    assert polygamma(Symbol('a', positive=True), Symbol('b', positive=True)).is_real is True
+    assert polygamma(0, I).is_real is None
+
+    assert str(polygamma(pi, 3).evalf(n=10)) == "0.1169314564"
+    assert str(polygamma(2.3, 1.0).evalf(n=10)) == "-3.003302909"
+    assert str(polygamma(-1, 1).evalf(n=10)) == "-0.9189385332" # not zero
+    assert str(polygamma(I, 1).evalf(n=10)) == "-3.109856569 + 1.89089016*I"
+    assert str(polygamma(1, I).evalf(n=10)) == "-0.5369999034 - 0.7942335428*I"
+    assert str(polygamma(I, I).evalf(n=10)) == "6.332362889 + 45.92828268*I"
+
+
+def test_polygamma_expand_func():
+    assert polygamma(0, x).expand(func=True) == polygamma(0, x)
+    assert polygamma(0, 2*x).expand(func=True) == \
+        polygamma(0, x)/2 + polygamma(0, S.Half + x)/2 + log(2)
+    assert polygamma(1, 2*x).expand(func=True) == \
+        polygamma(1, x)/4 + polygamma(1, S.Half + x)/4
+    assert polygamma(2, x).expand(func=True) == \
+        polygamma(2, x)
+    assert polygamma(0, -1 + x).expand(func=True) == \
+        polygamma(0, x) - 1/(x - 1)
+    assert polygamma(0, 1 + x).expand(func=True) == \
+        1/x + polygamma(0, x )
+    assert polygamma(0, 2 + x).expand(func=True) == \
+        1/x + 1/(1 + x) + polygamma(0, x)
+    assert polygamma(0, 3 + x).expand(func=True) == \
+        polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x)
+    assert polygamma(0, 4 + x).expand(func=True) == \
+        polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x) + 1/(3 + x)
+    assert polygamma(1, 1 + x).expand(func=True) == \
+        polygamma(1, x) - 1/x**2
+    assert polygamma(1, 2 + x).expand(func=True, multinomial=False) == \
+        polygamma(1, x) - 1/x**2 - 1/(1 + x)**2
+    assert polygamma(1, 3 + x).expand(func=True, multinomial=False) == \
+        polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - 1/(2 + x)**2
+    assert polygamma(1, 4 + x).expand(func=True, multinomial=False) == \
+        polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - \
+        1/(2 + x)**2 - 1/(3 + x)**2
+    assert polygamma(0, x + y).expand(func=True) == \
+        polygamma(0, x + y)
+    assert polygamma(1, x + y).expand(func=True) == \
+        polygamma(1, x + y)
+    assert polygamma(1, 3 + 4*x + y).expand(func=True, multinomial=False) == \
+        polygamma(1, y + 4*x) - 1/(y + 4*x)**2 - \
+        1/(1 + y + 4*x)**2 - 1/(2 + y + 4*x)**2
+    assert polygamma(3, 3 + 4*x + y).expand(func=True, multinomial=False) == \
+        polygamma(3, y + 4*x) - 6/(y + 4*x)**4 - \
+        6/(1 + y + 4*x)**4 - 6/(2 + y + 4*x)**4
+    assert polygamma(3, 4*x + y + 1).expand(func=True, multinomial=False) == \
+        polygamma(3, y + 4*x) - 6/(y + 4*x)**4
+    e = polygamma(3, 4*x + y + Rational(3, 2))
+    assert e.expand(func=True) == e
+    e = polygamma(3, x + y + Rational(3, 4))
+    assert e.expand(func=True, basic=False) == e
+
+    assert polygamma(-1, x, evaluate=False).expand(func=True) == \
+        loggamma(x) - log(pi)/2 - log(2)/2
+    p2 = polygamma(-2, x).expand(func=True) + x**2/2 - x/2 + S(1)/12
+    assert isinstance(p2, Subs)
+    assert p2.point == (-1,)
+
+
+def test_digamma():
+    assert digamma(nan) == nan
+
+    assert digamma(oo) == oo
+    assert digamma(-oo) == oo
+    assert digamma(I*oo) == oo
+    assert digamma(-I*oo) == oo
+
+    assert digamma(-9) == zoo
+
+    assert digamma(-9) == zoo
+    assert digamma(-1) == zoo
+
+    assert digamma(0) == zoo
+
+    assert digamma(1) == -EulerGamma
+    assert digamma(7) == Rational(49, 20) - EulerGamma
+
+    def t(m, n):
+        x = S(m)/n
+        r = digamma(x)
+        if r.has(digamma):
+            return False
+        return abs(digamma(x.n()).n() - r.n()).n() < 1e-10
+    assert t(1, 2)
+    assert t(3, 2)
+    assert t(-1, 2)
+    assert t(1, 4)
+    assert t(-3, 4)
+    assert t(1, 3)
+    assert t(4, 3)
+    assert t(3, 4)
+    assert t(2, 3)
+    assert t(123, 5)
+
+    assert digamma(x).rewrite(zeta) == polygamma(0, x)
+
+    assert digamma(x).rewrite(harmonic) == harmonic(x - 1) - EulerGamma
+
+    assert digamma(I).is_real is None
+
+    assert digamma(x,evaluate=False).fdiff() == polygamma(1, x)
+
+    assert digamma(x,evaluate=False).is_real is None
+
+    assert digamma(x,evaluate=False).is_positive is None
+
+    assert digamma(x,evaluate=False).is_negative is None
+
+    assert digamma(x,evaluate=False).rewrite(polygamma) == polygamma(0, x)
+
+
+def test_digamma_expand_func():
+    assert digamma(x).expand(func=True) == polygamma(0, x)
+    assert digamma(2*x).expand(func=True) == \
+        polygamma(0, x)/2 + polygamma(0, Rational(1, 2) + x)/2 + log(2)
+    assert digamma(-1 + x).expand(func=True) == \
+        polygamma(0, x) - 1/(x - 1)
+    assert digamma(1 + x).expand(func=True) == \
+        1/x + polygamma(0, x )
+    assert digamma(2 + x).expand(func=True) == \
+        1/x + 1/(1 + x) + polygamma(0, x)
+    assert digamma(3 + x).expand(func=True) == \
+        polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x)
+    assert digamma(4 + x).expand(func=True) == \
+        polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x) + 1/(3 + x)
+    assert digamma(x + y).expand(func=True) == \
+        polygamma(0, x + y)
+
+def test_trigamma():
+    assert trigamma(nan) == nan
+
+    assert trigamma(oo) == 0
+
+    assert trigamma(1) == pi**2/6
+    assert trigamma(2) == pi**2/6 - 1
+    assert trigamma(3) == pi**2/6 - Rational(5, 4)
+
+    assert trigamma(x, evaluate=False).rewrite(zeta) == zeta(2, x)
+    assert trigamma(x, evaluate=False).rewrite(harmonic) == \
+        trigamma(x).rewrite(polygamma).rewrite(harmonic)
+
+    assert trigamma(x,evaluate=False).fdiff() == polygamma(2, x)
+
+    assert trigamma(x,evaluate=False).is_real is None
+
+    assert trigamma(x,evaluate=False).is_positive is None
+
+    assert trigamma(x,evaluate=False).is_negative is None
+
+    assert trigamma(x,evaluate=False).rewrite(polygamma) == polygamma(1, x)
+
+def test_trigamma_expand_func():
+    assert trigamma(2*x).expand(func=True) == \
+        polygamma(1, x)/4 + polygamma(1, Rational(1, 2) + x)/4
+    assert trigamma(1 + x).expand(func=True) == \
+        polygamma(1, x) - 1/x**2
+    assert trigamma(2 + x).expand(func=True, multinomial=False) == \
+        polygamma(1, x) - 1/x**2 - 1/(1 + x)**2
+    assert trigamma(3 + x).expand(func=True, multinomial=False) == \
+        polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - 1/(2 + x)**2
+    assert trigamma(4 + x).expand(func=True, multinomial=False) == \
+        polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - \
+        1/(2 + x)**2 - 1/(3 + x)**2
+    assert trigamma(x + y).expand(func=True) == \
+        polygamma(1, x + y)
+    assert trigamma(3 + 4*x + y).expand(func=True, multinomial=False) == \
+        polygamma(1, y + 4*x) - 1/(y + 4*x)**2 - \
+        1/(1 + y + 4*x)**2 - 1/(2 + y + 4*x)**2
+
+def test_loggamma():
+    raises(TypeError, lambda: loggamma(2, 3))
+    raises(ArgumentIndexError, lambda: loggamma(x).fdiff(2))
+
+    assert loggamma(-1) is oo
+    assert loggamma(-2) is oo
+    assert loggamma(0) is oo
+    assert loggamma(1) == 0
+    assert loggamma(2) == 0
+    assert loggamma(3) == log(2)
+    assert loggamma(4) == log(6)
+
+    n = Symbol("n", integer=True, positive=True)
+    assert loggamma(n) == log(gamma(n))
+    assert loggamma(-n) is oo
+    assert loggamma(n/2) == log(2**(-n + 1)*sqrt(pi)*gamma(n)/gamma(n/2 + S.Half))
+
+    assert loggamma(oo) is oo
+    assert loggamma(-oo) is zoo
+    assert loggamma(I*oo) is zoo
+    assert loggamma(-I*oo) is zoo
+    assert loggamma(zoo) is zoo
+    assert loggamma(nan) is nan
+
+    L = loggamma(Rational(16, 3))
+    E = -5*log(3) + loggamma(Rational(1, 3)) + log(4) + log(7) + log(10) + log(13)
+    assert expand_func(L).doit() == E
+    assert L.n() == E.n()
+
+    L = loggamma(Rational(19, 4))
+    E = -4*log(4) + loggamma(Rational(3, 4)) + log(3) + log(7) + log(11) + log(15)
+    assert expand_func(L).doit() == E
+    assert L.n() == E.n()
+
+    L = loggamma(Rational(23, 7))
+    E = -3*log(7) + log(2) + loggamma(Rational(2, 7)) + log(9) + log(16)
+    assert expand_func(L).doit() == E
+    assert L.n() == E.n()
+
+    L = loggamma(Rational(19, 4) - 7)
+    E = -log(9) - log(5) + loggamma(Rational(3, 4)) + 3*log(4) - 3*I*pi
+    assert expand_func(L).doit() == E
+    assert L.n() == E.n()
+
+    L = loggamma(Rational(23, 7) - 6)
+    E = -log(19) - log(12) - log(5) + loggamma(Rational(2, 7)) + 3*log(7) - 3*I*pi
+    assert expand_func(L).doit() == E
+    assert L.n() == E.n()
+
+    assert loggamma(x).diff(x) == polygamma(0, x)
+    s1 = loggamma(1/(x + sin(x)) + cos(x)).nseries(x, n=4)
+    s2 = (-log(2*x) - 1)/(2*x) - log(x/pi)/2 + (4 - log(2*x))*x/24 + O(x**2) + \
+        log(x)*x**2/2
+    assert (s1 - s2).expand(force=True).removeO() == 0
+    s1 = loggamma(1/x).series(x)
+    s2 = (1/x - S.Half)*log(1/x) - 1/x + log(2*pi)/2 + \
+        x/12 - x**3/360 + x**5/1260 + O(x**7)
+    assert ((s1 - s2).expand(force=True)).removeO() == 0
+
+    assert loggamma(x).rewrite('intractable') == log(gamma(x))
+
+    s1 = loggamma(x).series(x).cancel()
+    assert s1 == -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + \
+        pi**4*x**4/360 + x**5*polygamma(4, 1)/120 + O(x**6)
+    assert s1 == loggamma(x).rewrite('intractable').series(x).cancel()
+
+    assert conjugate(loggamma(x)) == loggamma(conjugate(x))
+    assert conjugate(loggamma(0)) is oo
+    assert conjugate(loggamma(1)) == loggamma(conjugate(1))
+    assert conjugate(loggamma(-oo)) == conjugate(zoo)
+
+    assert loggamma(Symbol('v', positive=True)).is_real is True
+    assert loggamma(Symbol('v', zero=True)).is_real is False
+    assert loggamma(Symbol('v', negative=True)).is_real is False
+    assert loggamma(Symbol('v', nonpositive=True)).is_real is False
+    assert loggamma(Symbol('v', nonnegative=True)).is_real is None
+    assert loggamma(Symbol('v', imaginary=True)).is_real is None
+    assert loggamma(Symbol('v', real=True)).is_real is None
+    assert loggamma(Symbol('v')).is_real is None
+
+    assert loggamma(S.Half).is_real is True
+    assert loggamma(0).is_real is False
+    assert loggamma(Rational(-1, 2)).is_real is False
+    assert loggamma(I).is_real is None
+    assert loggamma(2 + 3*I).is_real is None
+
+    def tN(N, M):
+        assert loggamma(1/x)._eval_nseries(x, n=N).getn() == M
+    tN(0, 0)
+    tN(1, 1)
+    tN(2, 2)
+    tN(3, 3)
+    tN(4, 4)
+    tN(5, 5)
+
+
+def test_polygamma_expansion():
+    # A. & S., pa. 259 and 260
+    assert polygamma(0, 1/x).nseries(x, n=3) == \
+        -log(x) - x/2 - x**2/12 + O(x**3)
+    assert polygamma(1, 1/x).series(x, n=5) == \
+        x + x**2/2 + x**3/6 + O(x**5)
+    assert polygamma(3, 1/x).nseries(x, n=11) == \
+        2*x**3 + 3*x**4 + 2*x**5 - x**7 + 4*x**9/3 + O(x**11)
+
+
+def test_polygamma_leading_term():
+     expr = -log(1/x) + polygamma(0, 1 + 1/x) + S.EulerGamma
+     assert expr.as_leading_term(x, logx=-y) == S.EulerGamma
+
+
+def test_issue_8657():
+    n = Symbol('n', negative=True, integer=True)
+    m = Symbol('m', integer=True)
+    o = Symbol('o', positive=True)
+    p = Symbol('p', negative=True, integer=False)
+    assert gamma(n).is_real is False
+    assert gamma(m).is_real is None
+    assert gamma(o).is_real is True
+    assert gamma(p).is_real is True
+    assert gamma(w).is_real is None
+
+
+def test_issue_8524():
+    x = Symbol('x', positive=True)
+    y = Symbol('y', negative=True)
+    z = Symbol('z', positive=False)
+    p = Symbol('p', negative=False)
+    q = Symbol('q', integer=True)
+    r = Symbol('r', integer=False)
+    e = Symbol('e', even=True, negative=True)
+    assert gamma(x).is_positive is True
+    assert gamma(y).is_positive is None
+    assert gamma(z).is_positive is None
+    assert gamma(p).is_positive is None
+    assert gamma(q).is_positive is None
+    assert gamma(r).is_positive is None
+    assert gamma(e + S.Half).is_positive is True
+    assert gamma(e - S.Half).is_positive is False
+
+def test_issue_14450():
+    assert uppergamma(Rational(3, 8), x).evalf() == uppergamma(Rational(3, 8), x)
+    assert lowergamma(x, Rational(3, 8)).evalf() == lowergamma(x, Rational(3, 8))
+    # some values from Wolfram Alpha for comparison
+    assert abs(uppergamma(Rational(3, 8), 2).evalf() - 0.07105675881) < 1e-9
+    assert abs(lowergamma(Rational(3, 8), 2).evalf() - 2.2993794256) < 1e-9
+
+def test_issue_14528():
+    k = Symbol('k', integer=True, nonpositive=True)
+    assert isinstance(gamma(k), gamma)
+
+def test_multigamma():
+    from sympy.concrete.products import Product
+    p = Symbol('p')
+    _k = Dummy('_k')
+
+    assert multigamma(x, p).dummy_eq(pi**(p*(p - 1)/4)*\
+        Product(gamma(x + (1 - _k)/2), (_k, 1, p)))
+
+    assert conjugate(multigamma(x, p)).dummy_eq(pi**((conjugate(p) - 1)*\
+        conjugate(p)/4)*Product(gamma(conjugate(x) + (1-conjugate(_k))/2), (_k, 1, p)))
+    assert conjugate(multigamma(x, 1)) == gamma(conjugate(x))
+
+    p = Symbol('p', positive=True)
+    assert conjugate(multigamma(x, p)).dummy_eq(pi**((p - 1)*p/4)*\
+        Product(gamma(conjugate(x) + (1-conjugate(_k))/2), (_k, 1, p)))
+
+    assert multigamma(nan, 1) is nan
+    assert multigamma(oo, 1).doit() is oo
+
+    assert multigamma(1, 1) == 1
+    assert multigamma(2, 1) == 1
+    assert multigamma(3, 1) == 2
+
+    assert multigamma(102, 1) == factorial(101)
+    assert multigamma(S.Half, 1) == sqrt(pi)
+
+    assert multigamma(1, 2) == pi
+    assert multigamma(2, 2) == pi/2
+
+    assert multigamma(1, 3) is zoo
+    assert multigamma(2, 3) == pi**2/2
+    assert multigamma(3, 3) == 3*pi**2/2
+
+    assert multigamma(x, 1).diff(x) == gamma(x)*polygamma(0, x)
+    assert multigamma(x, 2).diff(x) == sqrt(pi)*gamma(x)*gamma(x - S.Half)*\
+        polygamma(0, x) + sqrt(pi)*gamma(x)*gamma(x - S.Half)*polygamma(0, x - S.Half)
+
+    assert multigamma(x - 1, 1).expand(func=True) == gamma(x)/(x - 1)
+    assert multigamma(x + 2, 1).expand(func=True, mul=False) == x*(x + 1)*\
+        gamma(x)
+    assert multigamma(x - 1, 2).expand(func=True) == sqrt(pi)*gamma(x)*\
+        gamma(x + S.Half)/(x**3 - 3*x**2 + x*Rational(11, 4) - Rational(3, 4))
+    assert multigamma(x - 1, 3).expand(func=True) == pi**Rational(3, 2)*gamma(x)**2*\
+        gamma(x + S.Half)/(x**5 - 6*x**4 + 55*x**3/4 - 15*x**2 + x*Rational(31, 4) - Rational(3, 2))
+
+    assert multigamma(n, 1).rewrite(factorial) == factorial(n - 1)
+    assert multigamma(n, 2).rewrite(factorial) == sqrt(pi)*\
+        factorial(n - Rational(3, 2))*factorial(n - 1)
+    assert multigamma(n, 3).rewrite(factorial) == pi**Rational(3, 2)*\
+        factorial(n - 2)*factorial(n - Rational(3, 2))*factorial(n - 1)
+
+    assert multigamma(Rational(-1, 2), 3, evaluate=False).is_real == False
+    assert multigamma(S.Half, 3, evaluate=False).is_real == False
+    assert multigamma(0, 1, evaluate=False).is_real == False
+    assert multigamma(1, 3, evaluate=False).is_real == False
+    assert multigamma(-1.0, 3, evaluate=False).is_real == False
+    assert multigamma(0.7, 3, evaluate=False).is_real == True
+    assert multigamma(3, 3, evaluate=False).is_real == True
+
+def test_gamma_as_leading_term():
+    assert gamma(x).as_leading_term(x) == 1/x
+    assert gamma(2 + x).as_leading_term(x) == S(1)
+    assert gamma(cos(x)).as_leading_term(x) == S(1)
+    assert gamma(sin(x)).as_leading_term(x) == 1/x

rl/Lib/site-packages/sympy/functions/special/tests/test_hyper.py

@@ -0,0 +1,403 @@
+from sympy.core.containers import Tuple
+from sympy.core.function import Derivative
+from sympy.core.numbers import (I, Rational, oo, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import cos
+from sympy.functions.special.gamma_functions import gamma
+from sympy.functions.special.hyper import (appellf1, hyper, meijerg)
+from sympy.series.order import O
+from sympy.abc import x, z, k
+from sympy.series.limits import limit
+from sympy.testing.pytest import raises, slow
+from sympy.core.random import (
+    random_complex_number as randcplx,
+    verify_numerically as tn,
+    test_derivative_numerically as td)
+
+
+def test_TupleParametersBase():
+    # test that our implementation of the chain rule works
+    p = hyper((), (), z**2)
+    assert p.diff(z) == p*2*z
+
+
+def test_hyper():
+    raises(TypeError, lambda: hyper(1, 2, z))
+
+    assert hyper((2, 1), (1,), z) == hyper(Tuple(1, 2), Tuple(1), z)
+    assert hyper((2, 1, 2), (1, 2, 1, 3), z) == hyper((2,), (1, 3), z)
+    u = hyper((2, 1, 2), (1, 2, 1, 3), z, evaluate=False)
+    assert u.ap == Tuple(1, 2, 2)
+    assert u.bq == Tuple(1, 1, 2, 3)
+
+    h = hyper((1, 2), (3, 4, 5), z)
+    assert h.ap == Tuple(1, 2)
+    assert h.bq == Tuple(3, 4, 5)
+    assert h.argument == z
+    assert h.is_commutative is True
+    h = hyper((2, 1), (4, 3, 5), z)
+    assert h.ap == Tuple(1, 2)
+    assert h.bq == Tuple(3, 4, 5)
+    assert h.argument == z
+    assert h.is_commutative is True
+
+    # just a few checks to make sure that all arguments go where they should
+    assert tn(hyper(Tuple(), Tuple(), z), exp(z), z)
+    assert tn(z*hyper((1, 1), Tuple(2), -z), log(1 + z), z)
+
+    # differentiation
+    h = hyper(
+        (randcplx(), randcplx(), randcplx()), (randcplx(), randcplx()), z)
+    assert td(h, z)
+
+    a1, a2, b1, b2, b3 = symbols('a1:3, b1:4')
+    assert hyper((a1, a2), (b1, b2, b3), z).diff(z) == \
+        a1*a2/(b1*b2*b3) * hyper((a1 + 1, a2 + 1), (b1 + 1, b2 + 1, b3 + 1), z)
+
+    # differentiation wrt parameters is not supported
+    assert hyper([z], [], z).diff(z) == Derivative(hyper([z], [], z), z)
+
+    # hyper is unbranched wrt parameters
+    from sympy.functions.elementary.complexes import polar_lift
+    assert hyper([polar_lift(z)], [polar_lift(k)], polar_lift(x)) == \
+        hyper([z], [k], polar_lift(x))
+
+    # hyper does not automatically evaluate anyway, but the test is to make
+    # sure that the evaluate keyword is accepted
+    assert hyper((1, 2), (1,), z, evaluate=False).func is hyper
+
+
+def test_expand_func():
+    # evaluation at 1 of Gauss' hypergeometric function:
+    from sympy.abc import a, b, c
+    from sympy.core.function import expand_func
+    a1, b1, c1 = randcplx(), randcplx(), randcplx() + 5
+    assert expand_func(hyper([a, b], [c], 1)) == \
+        gamma(c)*gamma(-a - b + c)/(gamma(-a + c)*gamma(-b + c))
+    assert abs(expand_func(hyper([a1, b1], [c1], 1)).n()
+               - hyper([a1, b1], [c1], 1).n()) < 1e-10
+
+    # hyperexpand wrapper for hyper:
+    assert expand_func(hyper([], [], z)) == exp(z)
+    assert expand_func(hyper([1, 2, 3], [], z)) == hyper([1, 2, 3], [], z)
+    assert expand_func(meijerg([[1, 1], []], [[1], [0]], z)) == log(z + 1)
+    assert expand_func(meijerg([[1, 1], []], [[], []], z)) == \
+        meijerg([[1, 1], []], [[], []], z)
+
+
+def replace_dummy(expr, sym):
+    from sympy.core.symbol import Dummy
+    dum = expr.atoms(Dummy)
+    if not dum:
+        return expr
+    assert len(dum) == 1
+    return expr.xreplace({dum.pop(): sym})
+
+
+def test_hyper_rewrite_sum():
+    from sympy.concrete.summations import Sum
+    from sympy.core.symbol import Dummy
+    from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial)
+    _k = Dummy("k")
+    assert replace_dummy(hyper((1, 2), (1, 3), x).rewrite(Sum), _k) == \
+        Sum(x**_k / factorial(_k) * RisingFactorial(2, _k) /
+            RisingFactorial(3, _k), (_k, 0, oo))
+
+    assert hyper((1, 2, 3), (-1, 3), z).rewrite(Sum) == \
+        hyper((1, 2, 3), (-1, 3), z)
+
+
+def test_radius_of_convergence():
+    assert hyper((1, 2), [3], z).radius_of_convergence == 1
+    assert hyper((1, 2), [3, 4], z).radius_of_convergence is oo
+    assert hyper((1, 2, 3), [4], z).radius_of_convergence == 0
+    assert hyper((0, 1, 2), [4], z).radius_of_convergence is oo
+    assert hyper((-1, 1, 2), [-4], z).radius_of_convergence == 0
+    assert hyper((-1, -2, 2), [-1], z).radius_of_convergence is oo
+    assert hyper((-1, 2), [-1, -2], z).radius_of_convergence == 0
+    assert hyper([-1, 1, 3], [-2, 2], z).radius_of_convergence == 1
+    assert hyper([-1, 1], [-2, 2], z).radius_of_convergence is oo
+    assert hyper([-1, 1, 3], [-2], z).radius_of_convergence == 0
+    assert hyper((-1, 2, 3, 4), [], z).radius_of_convergence is oo
+
+    assert hyper([1, 1], [3], 1).convergence_statement == True
+    assert hyper([1, 1], [2], 1).convergence_statement == False
+    assert hyper([1, 1], [2], -1).convergence_statement == True
+    assert hyper([1, 1], [1], -1).convergence_statement == False
+
+
+def test_meijer():
+    raises(TypeError, lambda: meijerg(1, z))
+    raises(TypeError, lambda: meijerg(((1,), (2,)), (3,), (4,), z))
+
+    assert meijerg(((1, 2), (3,)), ((4,), (5,)), z) == \
+        meijerg(Tuple(1, 2), Tuple(3), Tuple(4), Tuple(5), z)
+
+    g = meijerg((1, 2), (3, 4, 5), (6, 7, 8, 9), (10, 11, 12, 13, 14), z)
+    assert g.an == Tuple(1, 2)
+    assert g.ap == Tuple(1, 2, 3, 4, 5)
+    assert g.aother == Tuple(3, 4, 5)
+    assert g.bm == Tuple(6, 7, 8, 9)
+    assert g.bq == Tuple(6, 7, 8, 9, 10, 11, 12, 13, 14)
+    assert g.bother == Tuple(10, 11, 12, 13, 14)
+    assert g.argument == z
+    assert g.nu == 75
+    assert g.delta == -1
+    assert g.is_commutative is True
+    assert g.is_number is False
+    #issue 13071
+    assert meijerg([[],[]], [[S.Half],[0]], 1).is_number is True
+
+    assert meijerg([1, 2], [3], [4], [5], z).delta == S.Half
+
+    # just a few checks to make sure that all arguments go where they should
+    assert tn(meijerg(Tuple(), Tuple(), Tuple(0), Tuple(), -z), exp(z), z)
+    assert tn(sqrt(pi)*meijerg(Tuple(), Tuple(),
+                               Tuple(0), Tuple(S.Half), z**2/4), cos(z), z)
+    assert tn(meijerg(Tuple(1, 1), Tuple(), Tuple(1), Tuple(0), z),
+              log(1 + z), z)
+
+    # test exceptions
+    raises(ValueError, lambda: meijerg(((3, 1), (2,)), ((oo,), (2, 0)), x))
+    raises(ValueError, lambda: meijerg(((3, 1), (2,)), ((1,), (2, 0)), x))
+
+    # differentiation
+    g = meijerg((randcplx(),), (randcplx() + 2*I,), Tuple(),
+                (randcplx(), randcplx()), z)
+    assert td(g, z)
+
+    g = meijerg(Tuple(), (randcplx(),), Tuple(),
+                (randcplx(), randcplx()), z)
+    assert td(g, z)
+
+    g = meijerg(Tuple(), Tuple(), Tuple(randcplx()),
+                Tuple(randcplx(), randcplx()), z)
+    assert td(g, z)
+
+    a1, a2, b1, b2, c1, c2, d1, d2 = symbols('a1:3, b1:3, c1:3, d1:3')
+    assert meijerg((a1, a2), (b1, b2), (c1, c2), (d1, d2), z).diff(z) == \
+        (meijerg((a1 - 1, a2), (b1, b2), (c1, c2), (d1, d2), z)
+         + (a1 - 1)*meijerg((a1, a2), (b1, b2), (c1, c2), (d1, d2), z))/z
+
+    assert meijerg([z, z], [], [], [], z).diff(z) == \
+        Derivative(meijerg([z, z], [], [], [], z), z)
+
+    # meijerg is unbranched wrt parameters
+    from sympy.functions.elementary.complexes import polar_lift as pl
+    assert meijerg([pl(a1)], [pl(a2)], [pl(b1)], [pl(b2)], pl(z)) == \
+        meijerg([a1], [a2], [b1], [b2], pl(z))
+
+    # integrand
+    from sympy.abc import a, b, c, d, s
+    assert meijerg([a], [b], [c], [d], z).integrand(s) == \
+        z**s*gamma(c - s)*gamma(-a + s + 1)/(gamma(b - s)*gamma(-d + s + 1))
+
+
+def test_meijerg_derivative():
+    assert meijerg([], [1, 1], [0, 0, x], [], z).diff(x) == \
+        log(z)*meijerg([], [1, 1], [0, 0, x], [], z) \
+        + 2*meijerg([], [1, 1, 1], [0, 0, x, 0], [], z)
+
+    y = randcplx()
+    a = 5  # mpmath chokes with non-real numbers, and Mod1 with floats
+    assert td(meijerg([x], [], [], [], y), x)
+    assert td(meijerg([x**2], [], [], [], y), x)
+    assert td(meijerg([], [x], [], [], y), x)
+    assert td(meijerg([], [], [x], [], y), x)
+    assert td(meijerg([], [], [], [x], y), x)
+    assert td(meijerg([x], [a], [a + 1], [], y), x)
+    assert td(meijerg([x], [a + 1], [a], [], y), x)
+    assert td(meijerg([x, a], [], [], [a + 1], y), x)
+    assert td(meijerg([x, a + 1], [], [], [a], y), x)
+    b = Rational(3, 2)
+    assert td(meijerg([a + 2], [b], [b - 3, x], [a], y), x)
+
+
+def test_meijerg_period():
+    assert meijerg([], [1], [0], [], x).get_period() == 2*pi
+    assert meijerg([1], [], [], [0], x).get_period() == 2*pi
+    assert meijerg([], [], [0], [], x).get_period() == 2*pi  # exp(x)
+    assert meijerg(
+        [], [], [0], [S.Half], x).get_period() == 2*pi  # cos(sqrt(x))
+    assert meijerg(
+        [], [], [S.Half], [0], x).get_period() == 4*pi  # sin(sqrt(x))
+    assert meijerg([1, 1], [], [1], [0], x).get_period() is oo  # log(1 + x)
+
+
+def test_hyper_unpolarify():
+    from sympy.functions.elementary.exponential import exp_polar
+    a = exp_polar(2*pi*I)*x
+    b = x
+    assert hyper([], [], a).argument == b
+    assert hyper([0], [], a).argument == a
+    assert hyper([0], [0], a).argument == b
+    assert hyper([0, 1], [0], a).argument == a
+    assert hyper([0, 1], [0], exp_polar(2*pi*I)).argument == 1
+
+
+@slow
+def test_hyperrep():
+    from sympy.functions.special.hyper import (HyperRep, HyperRep_atanh,
+        HyperRep_power1, HyperRep_power2, HyperRep_log1, HyperRep_asin1,
+        HyperRep_asin2, HyperRep_sqrts1, HyperRep_sqrts2, HyperRep_log2,
+        HyperRep_cosasin, HyperRep_sinasin)
+    # First test the base class works.
+    from sympy.functions.elementary.exponential import exp_polar
+    from sympy.functions.elementary.piecewise import Piecewise
+    a, b, c, d, z = symbols('a b c d z')
+
+    class myrep(HyperRep):
+        @classmethod
+        def _expr_small(cls, x):
+            return a
+
+        @classmethod
+        def _expr_small_minus(cls, x):
+            return b
+
+        @classmethod
+        def _expr_big(cls, x, n):
+            return c*n
+
+        @classmethod
+        def _expr_big_minus(cls, x, n):
+            return d*n
+    assert myrep(z).rewrite('nonrep') == Piecewise((0, abs(z) > 1), (a, True))
+    assert myrep(exp_polar(I*pi)*z).rewrite('nonrep') == \
+        Piecewise((0, abs(z) > 1), (b, True))
+    assert myrep(exp_polar(2*I*pi)*z).rewrite('nonrep') == \
+        Piecewise((c, abs(z) > 1), (a, True))
+    assert myrep(exp_polar(3*I*pi)*z).rewrite('nonrep') == \
+        Piecewise((d, abs(z) > 1), (b, True))
+    assert myrep(exp_polar(4*I*pi)*z).rewrite('nonrep') == \
+        Piecewise((2*c, abs(z) > 1), (a, True))
+    assert myrep(exp_polar(5*I*pi)*z).rewrite('nonrep') == \
+        Piecewise((2*d, abs(z) > 1), (b, True))
+    assert myrep(z).rewrite('nonrepsmall') == a
+    assert myrep(exp_polar(I*pi)*z).rewrite('nonrepsmall') == b
+
+    def t(func, hyp, z):
+        """ Test that func is a valid representation of hyp. """
+        # First test that func agrees with hyp for small z
+        if not tn(func.rewrite('nonrepsmall'), hyp, z,
+                  a=Rational(-1, 2), b=Rational(-1, 2), c=S.Half, d=S.Half):
+            return False
+        # Next check that the two small representations agree.
+        if not tn(
+            func.rewrite('nonrepsmall').subs(
+                z, exp_polar(I*pi)*z).replace(exp_polar, exp),
+            func.subs(z, exp_polar(I*pi)*z).rewrite('nonrepsmall'),
+                z, a=Rational(-1, 2), b=Rational(-1, 2), c=S.Half, d=S.Half):
+            return False
+        # Next check continuity along exp_polar(I*pi)*t
+        expr = func.subs(z, exp_polar(I*pi)*z).rewrite('nonrep')
+        if abs(expr.subs(z, 1 + 1e-15).n() - expr.subs(z, 1 - 1e-15).n()) > 1e-10:
+            return False
+        # Finally check continuity of the big reps.
+
+        def dosubs(func, a, b):
+            rv = func.subs(z, exp_polar(a)*z).rewrite('nonrep')
+            return rv.subs(z, exp_polar(b)*z).replace(exp_polar, exp)
+        for n in [0, 1, 2, 3, 4, -1, -2, -3, -4]:
+            expr1 = dosubs(func, 2*I*pi*n, I*pi/2)
+            expr2 = dosubs(func, 2*I*pi*n + I*pi, -I*pi/2)
+            if not tn(expr1, expr2, z):
+                return False
+            expr1 = dosubs(func, 2*I*pi*(n + 1), -I*pi/2)
+            expr2 = dosubs(func, 2*I*pi*n + I*pi, I*pi/2)
+            if not tn(expr1, expr2, z):
+                return False
+        return True
+
+    # Now test the various representatives.
+    a = Rational(1, 3)
+    assert t(HyperRep_atanh(z), hyper([S.Half, 1], [Rational(3, 2)], z), z)
+    assert t(HyperRep_power1(a, z), hyper([-a], [], z), z)
+    assert t(HyperRep_power2(a, z), hyper([a, a - S.Half], [2*a], z), z)
+    assert t(HyperRep_log1(z), -z*hyper([1, 1], [2], z), z)
+    assert t(HyperRep_asin1(z), hyper([S.Half, S.Half], [Rational(3, 2)], z), z)
+    assert t(HyperRep_asin2(z), hyper([1, 1], [Rational(3, 2)], z), z)
+    assert t(HyperRep_sqrts1(a, z), hyper([-a, S.Half - a], [S.Half], z), z)
+    assert t(HyperRep_sqrts2(a, z),
+             -2*z/(2*a + 1)*hyper([-a - S.Half, -a], [S.Half], z).diff(z), z)
+    assert t(HyperRep_log2(z), -z/4*hyper([Rational(3, 2), 1, 1], [2, 2], z), z)
+    assert t(HyperRep_cosasin(a, z), hyper([-a, a], [S.Half], z), z)
+    assert t(HyperRep_sinasin(a, z), 2*a*z*hyper([1 - a, 1 + a], [Rational(3, 2)], z), z)
+
+
+@slow
+def test_meijerg_eval():
+    from sympy.functions.elementary.exponential import exp_polar
+    from sympy.functions.special.bessel import besseli
+    from sympy.abc import l
+    a = randcplx()
+    arg = x*exp_polar(k*pi*I)
+    expr1 = pi*meijerg([[], [(a + 1)/2]], [[a/2], [-a/2, (a + 1)/2]], arg**2/4)
+    expr2 = besseli(a, arg)
+
+    # Test that the two expressions agree for all arguments.
+    for x_ in [0.5, 1.5]:
+        for k_ in [0.0, 0.1, 0.3, 0.5, 0.8, 1, 5.751, 15.3]:
+            assert abs((expr1 - expr2).n(subs={x: x_, k: k_})) < 1e-10
+            assert abs((expr1 - expr2).n(subs={x: x_, k: -k_})) < 1e-10
+
+    # Test continuity independently
+    eps = 1e-13
+    expr2 = expr1.subs(k, l)
+    for x_ in [0.5, 1.5]:
+        for k_ in [0.5, Rational(1, 3), 0.25, 0.75, Rational(2, 3), 1.0, 1.5]:
+            assert abs((expr1 - expr2).n(
+                       subs={x: x_, k: k_ + eps, l: k_ - eps})) < 1e-10
+            assert abs((expr1 - expr2).n(
+                       subs={x: x_, k: -k_ + eps, l: -k_ - eps})) < 1e-10
+
+    expr = (meijerg(((0.5,), ()), ((0.5, 0, 0.5), ()), exp_polar(-I*pi)/4)
+            + meijerg(((0.5,), ()), ((0.5, 0, 0.5), ()), exp_polar(I*pi)/4)) \
+        /(2*sqrt(pi))
+    assert (expr - pi/exp(1)).n(chop=True) == 0
+
+
+def test_limits():
+    k, x = symbols('k, x')
+    assert hyper((1,), (Rational(4, 3), Rational(5, 3)), k**2).series(k) == \
+           1 + 9*k**2/20 + 81*k**4/1120 + O(k**6) # issue 6350
+
+    # https://github.com/sympy/sympy/issues/11465
+    assert limit(1/hyper((1, ), (1, ), x), x, 0) == 1
+
+
+def test_appellf1():
+    a, b1, b2, c, x, y = symbols('a b1 b2 c x y')
+    assert appellf1(a, b2, b1, c, y, x) == appellf1(a, b1, b2, c, x, y)
+    assert appellf1(a, b1, b1, c, y, x) == appellf1(a, b1, b1, c, x, y)
+    assert appellf1(a, b1, b2, c, S.Zero, S.Zero) is S.One
+
+    f = appellf1(a, b1, b2, c, S.Zero, S.Zero, evaluate=False)
+    assert f.func is appellf1
+    assert f.doit() is S.One
+
+
+def test_derivative_appellf1():
+    from sympy.core.function import diff
+    a, b1, b2, c, x, y, z = symbols('a b1 b2 c x y z')
+    assert diff(appellf1(a, b1, b2, c, x, y), x) == a*b1*appellf1(a + 1, b2, b1 + 1, c + 1, y, x)/c
+    assert diff(appellf1(a, b1, b2, c, x, y), y) == a*b2*appellf1(a + 1, b1, b2 + 1, c + 1, x, y)/c
+    assert diff(appellf1(a, b1, b2, c, x, y), z) == 0
+    assert diff(appellf1(a, b1, b2, c, x, y), a) ==  Derivative(appellf1(a, b1, b2, c, x, y), a)
+
+
+def test_eval_nseries():
+    a1, b1, a2, b2 = symbols('a1 b1 a2 b2')
+    assert hyper((1,2), (1,2,3), x**2)._eval_nseries(x, 7, None) == \
+        1 + x**2/3 + x**4/24 + x**6/360 + O(x**7)
+    assert exp(x)._eval_nseries(x,7,None) == \
+        hyper((a1, b1), (a1, b1), x)._eval_nseries(x, 7, None)
+    assert hyper((a1, a2), (b1, b2), x)._eval_nseries(z, 7, None) ==\
+        hyper((a1, a2), (b1, b2), x) + O(z**7)
+    assert hyper((-S(1)/2, S(1)/2), (1,), 4*x/(x + 1)).nseries(x) == \
+        1 - x + x**2/4 - 3*x**3/4 - 15*x**4/64 - 93*x**5/64 + O(x**6)
+    assert (pi/2*hyper((-S(1)/2, S(1)/2), (1,), 4*x/(x + 1))).nseries(x) == \
+        pi/2 - pi*x/2 + pi*x**2/8 - 3*pi*x**3/8 - 15*pi*x**4/128 - 93*pi*x**5/128 + O(x**6)

rl/Lib/site-packages/sympy/functions/special/tests/test_mathieu.py

@@ -0,0 +1,29 @@
+from sympy.core.function import diff
+from sympy.functions.elementary.complexes import conjugate
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.functions.special.mathieu_functions import (mathieuc, mathieucprime, mathieus, mathieusprime)
+
+from sympy.abc import a, q, z
+
+
+def test_mathieus():
+    assert isinstance(mathieus(a, q, z), mathieus)
+    assert mathieus(a, 0, z) == sin(sqrt(a)*z)
+    assert conjugate(mathieus(a, q, z)) == mathieus(conjugate(a), conjugate(q), conjugate(z))
+    assert diff(mathieus(a, q, z), z) == mathieusprime(a, q, z)
+
+def test_mathieuc():
+    assert isinstance(mathieuc(a, q, z), mathieuc)
+    assert mathieuc(a, 0, z) == cos(sqrt(a)*z)
+    assert diff(mathieuc(a, q, z), z) == mathieucprime(a, q, z)
+
+def test_mathieusprime():
+    assert isinstance(mathieusprime(a, q, z), mathieusprime)
+    assert mathieusprime(a, 0, z) == sqrt(a)*cos(sqrt(a)*z)
+    assert diff(mathieusprime(a, q, z), z) == (-a + 2*q*cos(2*z))*mathieus(a, q, z)
+
+def test_mathieucprime():
+    assert isinstance(mathieucprime(a, q, z), mathieucprime)
+    assert mathieucprime(a, 0, z) == -sqrt(a)*sin(sqrt(a)*z)
+    assert diff(mathieucprime(a, q, z), z) == (-a + 2*q*cos(2*z))*mathieuc(a, q, z)

rl/Lib/site-packages/sympy/functions/special/tests/test_singularity_functions.py

@@ -0,0 +1,129 @@
+from sympy.core.function import (Derivative, diff)
+from sympy.core.numbers import (Float, I, nan, oo, pi)
+from sympy.core.relational import Eq
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.special.delta_functions import (DiracDelta, Heaviside)
+from sympy.functions.special.singularity_functions import SingularityFunction
+from sympy.series.order import O
+
+
+from sympy.core.expr import unchanged
+from sympy.core.function import ArgumentIndexError
+from sympy.testing.pytest import raises
+
+x, y, a, n = symbols('x y a n')
+
+
+def test_fdiff():
+    assert SingularityFunction(x, 4, 5).fdiff() == 5*SingularityFunction(x, 4, 4)
+    assert SingularityFunction(x, 4, -1).fdiff() == SingularityFunction(x, 4, -2)
+    assert SingularityFunction(x, 4, -2).fdiff() == SingularityFunction(x, 4, -3)
+    assert SingularityFunction(x, 4, -3).fdiff() == SingularityFunction(x, 4, -4)
+    assert SingularityFunction(x, 4, 0).fdiff() == SingularityFunction(x, 4, -1)
+
+    assert SingularityFunction(y, 6, 2).diff(y) == 2*SingularityFunction(y, 6, 1)
+    assert SingularityFunction(y, -4, -1).diff(y) == SingularityFunction(y, -4, -2)
+    assert SingularityFunction(y, 4, 0).diff(y) == SingularityFunction(y, 4, -1)
+    assert SingularityFunction(y, 4, 0).diff(y, 2) == SingularityFunction(y, 4, -2)
+
+    n = Symbol('n', positive=True)
+    assert SingularityFunction(x, a, n).fdiff() == n*SingularityFunction(x, a, n - 1)
+    assert SingularityFunction(y, a, n).diff(y) == n*SingularityFunction(y, a, n - 1)
+
+    expr_in = 4*SingularityFunction(x, a, n) + 3*SingularityFunction(x, a, -1) + -10*SingularityFunction(x, a, 0)
+    expr_out = n*4*SingularityFunction(x, a, n - 1) + 3*SingularityFunction(x, a, -2) - 10*SingularityFunction(x, a, -1)
+    assert diff(expr_in, x) == expr_out
+
+    assert SingularityFunction(x, -10, 5).diff(evaluate=False) == (
+        Derivative(SingularityFunction(x, -10, 5), x))
+
+    raises(ArgumentIndexError, lambda: SingularityFunction(x, 4, 5).fdiff(2))
+
+
+def test_eval():
+    assert SingularityFunction(x, a, n).func == SingularityFunction
+    assert unchanged(SingularityFunction, x, 5, n)
+    assert SingularityFunction(5, 3, 2) == 4
+    assert SingularityFunction(3, 5, 1) == 0
+    assert SingularityFunction(3, 3, 0) == 1
+    assert SingularityFunction(3, 3, 1) == 0
+    assert SingularityFunction(Symbol('z', zero=True), 0, 1) == 0  # like sin(z) == 0
+    assert SingularityFunction(4, 4, -1) is oo
+    assert SingularityFunction(4, 2, -1) == 0
+    assert SingularityFunction(4, 7, -1) == 0
+    assert SingularityFunction(5, 6, -2) == 0
+    assert SingularityFunction(4, 2, -2) == 0
+    assert SingularityFunction(4, 4, -2) is oo
+    assert SingularityFunction(4, 2, -3) == 0
+    assert SingularityFunction(8, 8, -3) is oo
+    assert SingularityFunction(4, 2, -4) == 0
+    assert SingularityFunction(8, 8, -4) is oo
+    assert (SingularityFunction(6.1, 4, 5)).evalf(5) == Float('40.841', '5')
+    assert SingularityFunction(6.1, pi, 2) == (-pi + 6.1)**2
+    assert SingularityFunction(x, a, nan) is nan
+    assert SingularityFunction(x, nan, 1) is nan
+    assert SingularityFunction(nan, a, n) is nan
+
+    raises(ValueError, lambda: SingularityFunction(x, a, I))
+    raises(ValueError, lambda: SingularityFunction(2*I, I, n))
+    raises(ValueError, lambda: SingularityFunction(x, a, -5))
+
+
+def test_leading_term():
+    l = Symbol('l', positive=True)
+    assert SingularityFunction(x, 3, 2).as_leading_term(x) == 0
+    assert SingularityFunction(x, -2, 1).as_leading_term(x) == 2
+    assert SingularityFunction(x, 0, 0).as_leading_term(x) == 1
+    assert SingularityFunction(x, 0, 0).as_leading_term(x, cdir=-1) == 0
+    assert SingularityFunction(x, 0, -1).as_leading_term(x) == 0
+    assert SingularityFunction(x, 0, -2).as_leading_term(x) == 0
+    assert SingularityFunction(x, 0, -3).as_leading_term(x) == 0
+    assert SingularityFunction(x, 0, -4).as_leading_term(x) == 0
+    assert (SingularityFunction(x + l, 0, 1)/2\
+        - SingularityFunction(x + l, l/2, 1)\
+        + SingularityFunction(x + l, l, 1)/2).as_leading_term(x) == -x/2
+
+
+def test_series():
+    l = Symbol('l', positive=True)
+    assert SingularityFunction(x, -3, 2).series(x) == x**2 + 6*x + 9
+    assert SingularityFunction(x, -2, 1).series(x) == x + 2
+    assert SingularityFunction(x, 0, 0).series(x) == 1
+    assert SingularityFunction(x, 0, 0).series(x, dir='-') == 0
+    assert SingularityFunction(x, 0, -1).series(x) == 0
+    assert SingularityFunction(x, 0, -2).series(x) == 0
+    assert SingularityFunction(x, 0, -3).series(x) == 0
+    assert SingularityFunction(x, 0, -4).series(x) == 0
+    assert (SingularityFunction(x + l, 0, 1)/2\
+        - SingularityFunction(x + l, l/2, 1)\
+        + SingularityFunction(x + l, l, 1)/2).nseries(x) == -x/2 + O(x**6)
+
+
+def test_rewrite():
+    assert SingularityFunction(x, 4, 5).rewrite(Piecewise) == (
+        Piecewise(((x - 4)**5, x - 4 >= 0), (0, True)))
+    assert SingularityFunction(x, -10, 0).rewrite(Piecewise) == (
+        Piecewise((1, x + 10 >= 0), (0, True)))
+    assert SingularityFunction(x, 2, -1).rewrite(Piecewise) == (
+        Piecewise((oo, Eq(x - 2, 0)), (0, True)))
+    assert SingularityFunction(x, 0, -2).rewrite(Piecewise) == (
+        Piecewise((oo, Eq(x, 0)), (0, True)))
+
+    n = Symbol('n', nonnegative=True)
+    p = SingularityFunction(x, a, n).rewrite(Piecewise)
+    assert p == (
+        Piecewise(((x - a)**n, x - a >= 0), (0, True)))
+    assert p.subs(x, a).subs(n, 0) == 1
+
+    expr_in = SingularityFunction(x, 4, 5) + SingularityFunction(x, -3, -1) - SingularityFunction(x, 0, -2)
+    expr_out = (x - 4)**5*Heaviside(x - 4, 1) + DiracDelta(x + 3) - DiracDelta(x, 1)
+    assert expr_in.rewrite(Heaviside) == expr_out
+    assert expr_in.rewrite(DiracDelta) == expr_out
+    assert expr_in.rewrite('HeavisideDiracDelta') == expr_out
+
+    expr_in = SingularityFunction(x, a, n) + SingularityFunction(x, a, -1) - SingularityFunction(x, a, -2)
+    expr_out = (x - a)**n*Heaviside(x - a, 1) + DiracDelta(x - a) + DiracDelta(a - x, 1)
+    assert expr_in.rewrite(Heaviside) == expr_out
+    assert expr_in.rewrite(DiracDelta) == expr_out
+    assert expr_in.rewrite('HeavisideDiracDelta') == expr_out

rl/Lib/site-packages/sympy/functions/special/tests/test_spec_polynomials.py

@@ -0,0 +1,475 @@
+from sympy.concrete.summations import Sum
+from sympy.core.function import (Derivative, diff)
+from sympy.core.numbers import (Rational, oo, pi, zoo)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol)
+from sympy.functions.combinatorial.factorials import (RisingFactorial, binomial, factorial)
+from sympy.functions.elementary.complexes import conjugate
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.integers import floor
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import cos
+from sympy.functions.special.gamma_functions import gamma
+from sympy.functions.special.hyper import hyper
+from sympy.functions.special.polynomials import (assoc_laguerre, assoc_legendre, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, gegenbauer, hermite, hermite_prob, jacobi, jacobi_normalized, laguerre, legendre)
+from sympy.polys.orthopolys import laguerre_poly
+from sympy.polys.polyroots import roots
+
+from sympy.core.expr import unchanged
+from sympy.core.function import ArgumentIndexError
+from sympy.testing.pytest import raises
+
+
+x = Symbol('x')
+
+
+def test_jacobi():
+    n = Symbol("n")
+    a = Symbol("a")
+    b = Symbol("b")
+
+    assert jacobi(0, a, b, x) == 1
+    assert jacobi(1, a, b, x) == a/2 - b/2 + x*(a/2 + b/2 + 1)
+
+    assert jacobi(n, a, a, x) == RisingFactorial(
+        a + 1, n)*gegenbauer(n, a + S.Half, x)/RisingFactorial(2*a + 1, n)
+    assert jacobi(n, a, -a, x) == ((-1)**a*(-x + 1)**(-a/2)*(x + 1)**(a/2)*assoc_legendre(n, a, x)*
+                                   factorial(-a + n)*gamma(a + n + 1)/(factorial(a + n)*gamma(n + 1)))
+    assert jacobi(n, -b, b, x) == ((-x + 1)**(b/2)*(x + 1)**(-b/2)*assoc_legendre(n, b, x)*
+                                   gamma(-b + n + 1)/gamma(n + 1))
+    assert jacobi(n, 0, 0, x) == legendre(n, x)
+    assert jacobi(n, S.Half, S.Half, x) == RisingFactorial(
+        Rational(3, 2), n)*chebyshevu(n, x)/factorial(n + 1)
+    assert jacobi(n, Rational(-1, 2), Rational(-1, 2), x) == RisingFactorial(
+        S.Half, n)*chebyshevt(n, x)/factorial(n)
+
+    X = jacobi(n, a, b, x)
+    assert isinstance(X, jacobi)
+
+    assert jacobi(n, a, b, -x) == (-1)**n*jacobi(n, b, a, x)
+    assert jacobi(n, a, b, 0) == 2**(-n)*gamma(a + n + 1)*hyper(
+        (-b - n, -n), (a + 1,), -1)/(factorial(n)*gamma(a + 1))
+    assert jacobi(n, a, b, 1) == RisingFactorial(a + 1, n)/factorial(n)
+
+    m = Symbol("m", positive=True)
+    assert jacobi(m, a, b, oo) == oo*RisingFactorial(a + b + m + 1, m)
+    assert unchanged(jacobi, n, a, b, oo)
+
+    assert conjugate(jacobi(m, a, b, x)) == \
+        jacobi(m, conjugate(a), conjugate(b), conjugate(x))
+
+    _k = Dummy('k')
+    assert diff(jacobi(n, a, b, x), n) == Derivative(jacobi(n, a, b, x), n)
+    assert diff(jacobi(n, a, b, x), a).dummy_eq(Sum((jacobi(n, a, b, x) +
+        (2*_k + a + b + 1)*RisingFactorial(_k + b + 1, -_k + n)*jacobi(_k, a,
+        b, x)/((-_k + n)*RisingFactorial(_k + a + b + 1, -_k + n)))/(_k + a
+        + b + n + 1), (_k, 0, n - 1)))
+    assert diff(jacobi(n, a, b, x), b).dummy_eq(Sum(((-1)**(-_k + n)*(2*_k +
+        a + b + 1)*RisingFactorial(_k + a + 1, -_k + n)*jacobi(_k, a, b, x)/
+        ((-_k + n)*RisingFactorial(_k + a + b + 1, -_k + n)) + jacobi(n, a,
+        b, x))/(_k + a + b + n + 1), (_k, 0, n - 1)))
+    assert diff(jacobi(n, a, b, x), x) == \
+        (a/2 + b/2 + n/2 + S.Half)*jacobi(n - 1, a + 1, b + 1, x)
+
+    assert jacobi_normalized(n, a, b, x) == \
+           (jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)
+                                    /((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1))))
+
+    raises(ValueError, lambda: jacobi(-2.1, a, b, x))
+    raises(ValueError, lambda: jacobi(Dummy(positive=True, integer=True), 1, 2, oo))
+
+    assert jacobi(n, a, b, x).rewrite(Sum).dummy_eq(Sum((S.Half - x/2)
+        **_k*RisingFactorial(-n, _k)*RisingFactorial(_k + a + 1, -_k + n)*
+        RisingFactorial(a + b + n + 1, _k)/factorial(_k), (_k, 0, n))/factorial(n))
+    assert jacobi(n, a, b, x).rewrite("polynomial").dummy_eq(Sum((S.Half - x/2)
+        **_k*RisingFactorial(-n, _k)*RisingFactorial(_k + a + 1, -_k + n)*
+        RisingFactorial(a + b + n + 1, _k)/factorial(_k), (_k, 0, n))/factorial(n))
+    raises(ArgumentIndexError, lambda: jacobi(n, a, b, x).fdiff(5))
+
+
+def test_gegenbauer():
+    n = Symbol("n")
+    a = Symbol("a")
+
+    assert gegenbauer(0, a, x) == 1
+    assert gegenbauer(1, a, x) == 2*a*x
+    assert gegenbauer(2, a, x) == -a + x**2*(2*a**2 + 2*a)
+    assert gegenbauer(3, a, x) == \
+        x**3*(4*a**3/3 + 4*a**2 + a*Rational(8, 3)) + x*(-2*a**2 - 2*a)
+
+    assert gegenbauer(-1, a, x) == 0
+    assert gegenbauer(n, S.Half, x) == legendre(n, x)
+    assert gegenbauer(n, 1, x) == chebyshevu(n, x)
+    assert gegenbauer(n, -1, x) == 0
+
+    X = gegenbauer(n, a, x)
+    assert isinstance(X, gegenbauer)
+
+    assert gegenbauer(n, a, -x) == (-1)**n*gegenbauer(n, a, x)
+    assert gegenbauer(n, a, 0) == 2**n*sqrt(pi) * \
+        gamma(a + n/2)/(gamma(a)*gamma(-n/2 + S.Half)*gamma(n + 1))
+    assert gegenbauer(n, a, 1) == gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))
+
+    assert gegenbauer(n, Rational(3, 4), -1) is zoo
+    assert gegenbauer(n, Rational(1, 4), -1) == (sqrt(2)*cos(pi*(n + S.One/4))*
+                      gamma(n + S.Half)/(sqrt(pi)*gamma(n + 1)))
+
+    m = Symbol("m", positive=True)
+    assert gegenbauer(m, a, oo) == oo*RisingFactorial(a, m)
+    assert unchanged(gegenbauer, n, a, oo)
+
+    assert conjugate(gegenbauer(n, a, x)) == gegenbauer(n, conjugate(a), conjugate(x))
+
+    _k = Dummy('k')
+
+    assert diff(gegenbauer(n, a, x), n) == Derivative(gegenbauer(n, a, x), n)
+    assert diff(gegenbauer(n, a, x), a).dummy_eq(Sum((2*(-1)**(-_k + n) + 2)*
+        (_k + a)*gegenbauer(_k, a, x)/((-_k + n)*(_k + 2*a + n)) + ((2*_k +
+        2)/((_k + 2*a)*(2*_k + 2*a + 1)) + 2/(_k + 2*a + n))*gegenbauer(n, a
+        , x), (_k, 0, n - 1)))
+    assert diff(gegenbauer(n, a, x), x) == 2*a*gegenbauer(n - 1, a + 1, x)
+
+    assert gegenbauer(n, a, x).rewrite(Sum).dummy_eq(
+        Sum((-1)**_k*(2*x)**(-2*_k + n)*RisingFactorial(a, -_k + n)
+        /(factorial(_k)*factorial(-2*_k + n)), (_k, 0, floor(n/2))))
+    assert gegenbauer(n, a, x).rewrite("polynomial").dummy_eq(
+        Sum((-1)**_k*(2*x)**(-2*_k + n)*RisingFactorial(a, -_k + n)
+        /(factorial(_k)*factorial(-2*_k + n)), (_k, 0, floor(n/2))))
+
+    raises(ArgumentIndexError, lambda: gegenbauer(n, a, x).fdiff(4))
+
+
+def test_legendre():
+    assert legendre(0, x) == 1
+    assert legendre(1, x) == x
+    assert legendre(2, x) == ((3*x**2 - 1)/2).expand()
+    assert legendre(3, x) == ((5*x**3 - 3*x)/2).expand()
+    assert legendre(4, x) == ((35*x**4 - 30*x**2 + 3)/8).expand()
+    assert legendre(5, x) == ((63*x**5 - 70*x**3 + 15*x)/8).expand()
+    assert legendre(6, x) == ((231*x**6 - 315*x**4 + 105*x**2 - 5)/16).expand()
+
+    assert legendre(10, -1) == 1
+    assert legendre(11, -1) == -1
+    assert legendre(10, 1) == 1
+    assert legendre(11, 1) == 1
+    assert legendre(10, 0) != 0
+    assert legendre(11, 0) == 0
+
+    assert legendre(-1, x) == 1
+    k = Symbol('k')
+    assert legendre(5 - k, x).subs(k, 2) == ((5*x**3 - 3*x)/2).expand()
+
+    assert roots(legendre(4, x), x) == {
+        sqrt(Rational(3, 7) - Rational(2, 35)*sqrt(30)): 1,
+        -sqrt(Rational(3, 7) - Rational(2, 35)*sqrt(30)): 1,
+        sqrt(Rational(3, 7) + Rational(2, 35)*sqrt(30)): 1,
+        -sqrt(Rational(3, 7) + Rational(2, 35)*sqrt(30)): 1,
+    }
+
+    n = Symbol("n")
+
+    X = legendre(n, x)
+    assert isinstance(X, legendre)
+    assert unchanged(legendre, n, x)
+
+    assert legendre(n, 0) == sqrt(pi)/(gamma(S.Half - n/2)*gamma(n/2 + 1))
+    assert legendre(n, 1) == 1
+    assert legendre(n, oo) is oo
+    assert legendre(-n, x) == legendre(n - 1, x)
+    assert legendre(n, -x) == (-1)**n*legendre(n, x)
+    assert unchanged(legendre, -n + k, x)
+
+    assert conjugate(legendre(n, x)) == legendre(n, conjugate(x))
+
+    assert diff(legendre(n, x), x) == \
+        n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)
+    assert diff(legendre(n, x), n) == Derivative(legendre(n, x), n)
+
+    _k = Dummy('k')
+    assert legendre(n, x).rewrite(Sum).dummy_eq(Sum((-1)**_k*(S.Half -
+            x/2)**_k*(x/2 + S.Half)**(-_k + n)*binomial(n, _k)**2, (_k, 0, n)))
+    assert legendre(n, x).rewrite("polynomial").dummy_eq(Sum((-1)**_k*(S.Half -
+            x/2)**_k*(x/2 + S.Half)**(-_k + n)*binomial(n, _k)**2, (_k, 0, n)))
+    raises(ArgumentIndexError, lambda: legendre(n, x).fdiff(1))
+    raises(ArgumentIndexError, lambda: legendre(n, x).fdiff(3))
+
+
+def test_assoc_legendre():
+    Plm = assoc_legendre
+    Q = sqrt(1 - x**2)
+
+    assert Plm(0, 0, x) == 1
+    assert Plm(1, 0, x) == x
+    assert Plm(1, 1, x) == -Q
+    assert Plm(2, 0, x) == (3*x**2 - 1)/2
+    assert Plm(2, 1, x) == -3*x*Q
+    assert Plm(2, 2, x) == 3*Q**2
+    assert Plm(3, 0, x) == (5*x**3 - 3*x)/2
+    assert Plm(3, 1, x).expand() == (( 3*(1 - 5*x**2)/2 ).expand() * Q).expand()
+    assert Plm(3, 2, x) == 15*x * Q**2
+    assert Plm(3, 3, x) == -15 * Q**3
+
+    # negative m
+    assert Plm(1, -1, x) == -Plm(1, 1, x)/2
+    assert Plm(2, -2, x) == Plm(2, 2, x)/24
+    assert Plm(2, -1, x) == -Plm(2, 1, x)/6
+    assert Plm(3, -3, x) == -Plm(3, 3, x)/720
+    assert Plm(3, -2, x) == Plm(3, 2, x)/120
+    assert Plm(3, -1, x) == -Plm(3, 1, x)/12
+
+    n = Symbol("n")
+    m = Symbol("m")
+    X = Plm(n, m, x)
+    assert isinstance(X, assoc_legendre)
+
+    assert Plm(n, 0, x) == legendre(n, x)
+    assert Plm(n, m, 0) == 2**m*sqrt(pi)/(gamma(-m/2 - n/2 +
+                           S.Half)*gamma(-m/2 + n/2 + 1))
+
+    assert diff(Plm(m, n, x), x) == (m*x*assoc_legendre(m, n, x) -
+                (m + n)*assoc_legendre(m - 1, n, x))/(x**2 - 1)
+
+    _k = Dummy('k')
+    assert Plm(m, n, x).rewrite(Sum).dummy_eq(
+            (1 - x**2)**(n/2)*Sum((-1)**_k*2**(-m)*x**(-2*_k + m - n)*factorial
+             (-2*_k + 2*m)/(factorial(_k)*factorial(-_k + m)*factorial(-2*_k + m
+              - n)), (_k, 0, floor(m/2 - n/2))))
+    assert Plm(m, n, x).rewrite("polynomial").dummy_eq(
+            (1 - x**2)**(n/2)*Sum((-1)**_k*2**(-m)*x**(-2*_k + m - n)*factorial
+             (-2*_k + 2*m)/(factorial(_k)*factorial(-_k + m)*factorial(-2*_k + m
+              - n)), (_k, 0, floor(m/2 - n/2))))
+    assert conjugate(assoc_legendre(n, m, x)) == \
+        assoc_legendre(n, conjugate(m), conjugate(x))
+    raises(ValueError, lambda: Plm(0, 1, x))
+    raises(ValueError, lambda: Plm(-1, 1, x))
+    raises(ArgumentIndexError, lambda: Plm(n, m, x).fdiff(1))
+    raises(ArgumentIndexError, lambda: Plm(n, m, x).fdiff(2))
+    raises(ArgumentIndexError, lambda: Plm(n, m, x).fdiff(4))
+
+
+def test_chebyshev():
+    assert chebyshevt(0, x) == 1
+    assert chebyshevt(1, x) == x
+    assert chebyshevt(2, x) == 2*x**2 - 1
+    assert chebyshevt(3, x) == 4*x**3 - 3*x
+
+    for n in range(1, 4):
+        for k in range(n):
+            z = chebyshevt_root(n, k)
+            assert chebyshevt(n, z) == 0
+        raises(ValueError, lambda: chebyshevt_root(n, n))
+
+    for n in range(1, 4):
+        for k in range(n):
+            z = chebyshevu_root(n, k)
+            assert chebyshevu(n, z) == 0
+        raises(ValueError, lambda: chebyshevu_root(n, n))
+
+    n = Symbol("n")
+    X = chebyshevt(n, x)
+    assert isinstance(X, chebyshevt)
+    assert unchanged(chebyshevt, n, x)
+    assert chebyshevt(n, -x) == (-1)**n*chebyshevt(n, x)
+    assert chebyshevt(-n, x) == chebyshevt(n, x)
+
+    assert chebyshevt(n, 0) == cos(pi*n/2)
+    assert chebyshevt(n, 1) == 1
+    assert chebyshevt(n, oo) is oo
+
+    assert conjugate(chebyshevt(n, x)) == chebyshevt(n, conjugate(x))
+
+    assert diff(chebyshevt(n, x), x) == n*chebyshevu(n - 1, x)
+
+    X = chebyshevu(n, x)
+    assert isinstance(X, chebyshevu)
+
+    y = Symbol('y')
+    assert chebyshevu(n, -x) == (-1)**n*chebyshevu(n, x)
+    assert chebyshevu(-n, x) == -chebyshevu(n - 2, x)
+    assert unchanged(chebyshevu, -n + y, x)
+
+    assert chebyshevu(n, 0) == cos(pi*n/2)
+    assert chebyshevu(n, 1) == n + 1
+    assert chebyshevu(n, oo) is oo
+
+    assert conjugate(chebyshevu(n, x)) == chebyshevu(n, conjugate(x))
+
+    assert diff(chebyshevu(n, x), x) == \
+        (-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)
+
+    _k = Dummy('k')
+    assert chebyshevt(n, x).rewrite(Sum).dummy_eq(Sum(x**(-2*_k + n)
+                    *(x**2 - 1)**_k*binomial(n, 2*_k), (_k, 0, floor(n/2))))
+    assert chebyshevt(n, x).rewrite("polynomial").dummy_eq(Sum(x**(-2*_k + n)
+                    *(x**2 - 1)**_k*binomial(n, 2*_k), (_k, 0, floor(n/2))))
+    assert chebyshevu(n, x).rewrite(Sum).dummy_eq(Sum((-1)**_k*(2*x)
+                    **(-2*_k + n)*factorial(-_k + n)/(factorial(_k)*
+                       factorial(-2*_k + n)), (_k, 0, floor(n/2))))
+    assert chebyshevu(n, x).rewrite("polynomial").dummy_eq(Sum((-1)**_k*(2*x)
+                    **(-2*_k + n)*factorial(-_k + n)/(factorial(_k)*
+                       factorial(-2*_k + n)), (_k, 0, floor(n/2))))
+    raises(ArgumentIndexError, lambda: chebyshevt(n, x).fdiff(1))
+    raises(ArgumentIndexError, lambda: chebyshevt(n, x).fdiff(3))
+    raises(ArgumentIndexError, lambda: chebyshevu(n, x).fdiff(1))
+    raises(ArgumentIndexError, lambda: chebyshevu(n, x).fdiff(3))
+
+
+def test_hermite():
+    assert hermite(0, x) == 1
+    assert hermite(1, x) == 2*x
+    assert hermite(2, x) == 4*x**2 - 2
+    assert hermite(3, x) == 8*x**3 - 12*x
+    assert hermite(4, x) == 16*x**4 - 48*x**2 + 12
+    assert hermite(6, x) == 64*x**6 - 480*x**4 + 720*x**2 - 120
+
+    n = Symbol("n")
+    assert unchanged(hermite, n, x)
+    assert hermite(n, -x) == (-1)**n*hermite(n, x)
+    assert unchanged(hermite, -n, x)
+
+    assert hermite(n, 0) == 2**n*sqrt(pi)/gamma(S.Half - n/2)
+    assert hermite(n, oo) is oo
+
+    assert conjugate(hermite(n, x)) == hermite(n, conjugate(x))
+
+    _k = Dummy('k')
+    assert hermite(n, x).rewrite(Sum).dummy_eq(factorial(n)*Sum((-1)
+        **_k*(2*x)**(-2*_k + n)/(factorial(_k)*factorial(-2*_k + n)), (_k,
+        0, floor(n/2))))
+    assert hermite(n, x).rewrite("polynomial").dummy_eq(factorial(n)*Sum((-1)
+        **_k*(2*x)**(-2*_k + n)/(factorial(_k)*factorial(-2*_k + n)), (_k,
+        0, floor(n/2))))
+
+    assert diff(hermite(n, x), x) == 2*n*hermite(n - 1, x)
+    assert diff(hermite(n, x), n) == Derivative(hermite(n, x), n)
+    raises(ArgumentIndexError, lambda: hermite(n, x).fdiff(3))
+
+    assert hermite(n, x).rewrite(hermite_prob) == \
+            sqrt(2)**n * hermite_prob(n, x*sqrt(2))
+
+
+def test_hermite_prob():
+    assert hermite_prob(0, x) == 1
+    assert hermite_prob(1, x) == x
+    assert hermite_prob(2, x) == x**2 - 1
+    assert hermite_prob(3, x) == x**3 - 3*x
+    assert hermite_prob(4, x) == x**4 - 6*x**2 + 3
+    assert hermite_prob(6, x) == x**6 - 15*x**4 + 45*x**2 - 15
+
+    n = Symbol("n")
+    assert unchanged(hermite_prob, n, x)
+    assert hermite_prob(n, -x) == (-1)**n*hermite_prob(n, x)
+    assert unchanged(hermite_prob, -n, x)
+
+    assert hermite_prob(n, 0) == sqrt(pi)/gamma(S.Half - n/2)
+    assert hermite_prob(n, oo) is oo
+
+    assert conjugate(hermite_prob(n, x)) == hermite_prob(n, conjugate(x))
+
+    _k = Dummy('k')
+    assert hermite_prob(n, x).rewrite(Sum).dummy_eq(factorial(n) *
+        Sum((-S.Half)**_k * x**(n-2*_k) / (factorial(_k) * factorial(n-2*_k)),
+        (_k, 0, floor(n/2))))
+    assert hermite_prob(n, x).rewrite("polynomial").dummy_eq(factorial(n) *
+        Sum((-S.Half)**_k * x**(n-2*_k) / (factorial(_k) * factorial(n-2*_k)),
+        (_k, 0, floor(n/2))))
+
+    assert diff(hermite_prob(n, x), x) == n*hermite_prob(n-1, x)
+    assert diff(hermite_prob(n, x), n) == Derivative(hermite_prob(n, x), n)
+    raises(ArgumentIndexError, lambda: hermite_prob(n, x).fdiff(3))
+
+    assert hermite_prob(n, x).rewrite(hermite) == \
+            sqrt(2)**(-n) * hermite(n, x/sqrt(2))
+
+
+def test_laguerre():
+    n = Symbol("n")
+    m = Symbol("m", negative=True)
+
+    # Laguerre polynomials:
+    assert laguerre(0, x) == 1
+    assert laguerre(1, x) == -x + 1
+    assert laguerre(2, x) == x**2/2 - 2*x + 1
+    assert laguerre(3, x) == -x**3/6 + 3*x**2/2 - 3*x + 1
+    assert laguerre(-2, x) == (x + 1)*exp(x)
+
+    X = laguerre(n, x)
+    assert isinstance(X, laguerre)
+
+    assert laguerre(n, 0) == 1
+    assert laguerre(n, oo) == (-1)**n*oo
+    assert laguerre(n, -oo) is oo
+
+    assert conjugate(laguerre(n, x)) == laguerre(n, conjugate(x))
+
+    _k = Dummy('k')
+
+    assert laguerre(n, x).rewrite(Sum).dummy_eq(
+        Sum(x**_k*RisingFactorial(-n, _k)/factorial(_k)**2, (_k, 0, n)))
+    assert laguerre(n, x).rewrite("polynomial").dummy_eq(
+        Sum(x**_k*RisingFactorial(-n, _k)/factorial(_k)**2, (_k, 0, n)))
+    assert laguerre(m, x).rewrite(Sum).dummy_eq(
+        exp(x)*Sum((-x)**_k*RisingFactorial(m + 1, _k)/factorial(_k)**2,
+            (_k, 0, -m - 1)))
+    assert laguerre(m, x).rewrite("polynomial").dummy_eq(
+        exp(x)*Sum((-x)**_k*RisingFactorial(m + 1, _k)/factorial(_k)**2,
+            (_k, 0, -m - 1)))
+
+    assert diff(laguerre(n, x), x) == -assoc_laguerre(n - 1, 1, x)
+
+    k = Symbol('k')
+    assert laguerre(-n, x) == exp(x)*laguerre(n - 1, -x)
+    assert laguerre(-3, x) == exp(x)*laguerre(2, -x)
+    assert unchanged(laguerre, -n + k, x)
+
+    raises(ValueError, lambda: laguerre(-2.1, x))
+    raises(ValueError, lambda: laguerre(Rational(5, 2), x))
+    raises(ArgumentIndexError, lambda: laguerre(n, x).fdiff(1))
+    raises(ArgumentIndexError, lambda: laguerre(n, x).fdiff(3))
+
+
+def test_assoc_laguerre():
+    n = Symbol("n")
+    m = Symbol("m")
+    alpha = Symbol("alpha")
+
+    # generalized Laguerre polynomials:
+    assert assoc_laguerre(0, alpha, x) == 1
+    assert assoc_laguerre(1, alpha, x) == -x + alpha + 1
+    assert assoc_laguerre(2, alpha, x).expand() == \
+        (x**2/2 - (alpha + 2)*x + (alpha + 2)*(alpha + 1)/2).expand()
+    assert assoc_laguerre(3, alpha, x).expand() == \
+        (-x**3/6 + (alpha + 3)*x**2/2 - (alpha + 2)*(alpha + 3)*x/2 +
+        (alpha + 1)*(alpha + 2)*(alpha + 3)/6).expand()
+
+    # Test the lowest 10 polynomials with laguerre_poly, to make sure it works:
+    for i in range(10):
+        assert assoc_laguerre(i, 0, x).expand() == laguerre_poly(i, x)
+
+    X = assoc_laguerre(n, m, x)
+    assert isinstance(X, assoc_laguerre)
+
+    assert assoc_laguerre(n, 0, x) == laguerre(n, x)
+    assert assoc_laguerre(n, alpha, 0) == binomial(alpha + n, alpha)
+    p = Symbol("p", positive=True)
+    assert assoc_laguerre(p, alpha, oo) == (-1)**p*oo
+    assert assoc_laguerre(p, alpha, -oo) is oo
+
+    assert diff(assoc_laguerre(n, alpha, x), x) == \
+        -assoc_laguerre(n - 1, alpha + 1, x)
+    _k = Dummy('k')
+    assert diff(assoc_laguerre(n, alpha, x), alpha).dummy_eq(
+        Sum(assoc_laguerre(_k, alpha, x)/(-alpha + n), (_k, 0, n - 1)))
+
+    assert conjugate(assoc_laguerre(n, alpha, x)) == \
+        assoc_laguerre(n, conjugate(alpha), conjugate(x))
+
+    assert assoc_laguerre(n, alpha, x).rewrite(Sum).dummy_eq(
+            gamma(alpha + n + 1)*Sum(x**_k*RisingFactorial(-n, _k)/
+            (factorial(_k)*gamma(_k + alpha + 1)), (_k, 0, n))/factorial(n))
+    assert assoc_laguerre(n, alpha, x).rewrite("polynomial").dummy_eq(
+            gamma(alpha + n + 1)*Sum(x**_k*RisingFactorial(-n, _k)/
+            (factorial(_k)*gamma(_k + alpha + 1)), (_k, 0, n))/factorial(n))
+    raises(ValueError, lambda: assoc_laguerre(-2.1, alpha, x))
+    raises(ArgumentIndexError, lambda: assoc_laguerre(n, alpha, x).fdiff(1))
+    raises(ArgumentIndexError, lambda: assoc_laguerre(n, alpha, x).fdiff(4))

rl/Lib/site-packages/sympy/functions/special/tests/test_spherical_harmonics.py

@@ -0,0 +1,66 @@
+from sympy.core.function import diff
+from sympy.core.numbers import (I, pi)
+from sympy.core.symbol import Symbol
+from sympy.functions.elementary.complexes import conjugate
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, cot, sin)
+from sympy.functions.special.spherical_harmonics import Ynm, Znm, Ynm_c
+
+
+def test_Ynm():
+    # https://en.wikipedia.org/wiki/Spherical_harmonics
+    th, ph = Symbol("theta", real=True), Symbol("phi", real=True)
+    from sympy.abc import n,m
+
+    assert Ynm(0, 0, th, ph).expand(func=True) == 1/(2*sqrt(pi))
+    assert Ynm(1, -1, th, ph) == -exp(-2*I*ph)*Ynm(1, 1, th, ph)
+    assert Ynm(1, -1, th, ph).expand(func=True) == sqrt(6)*sin(th)*exp(-I*ph)/(4*sqrt(pi))
+    assert Ynm(1, 0, th, ph).expand(func=True) == sqrt(3)*cos(th)/(2*sqrt(pi))
+    assert Ynm(1, 1, th, ph).expand(func=True) == -sqrt(6)*sin(th)*exp(I*ph)/(4*sqrt(pi))
+    assert Ynm(2, 0, th, ph).expand(func=True) == 3*sqrt(5)*cos(th)**2/(4*sqrt(pi)) - sqrt(5)/(4*sqrt(pi))
+    assert Ynm(2, 1, th, ph).expand(func=True) == -sqrt(30)*sin(th)*exp(I*ph)*cos(th)/(4*sqrt(pi))
+    assert Ynm(2, -2, th, ph).expand(func=True) == (-sqrt(30)*exp(-2*I*ph)*cos(th)**2/(8*sqrt(pi))
+                                                    + sqrt(30)*exp(-2*I*ph)/(8*sqrt(pi)))
+    assert Ynm(2, 2, th, ph).expand(func=True) == (-sqrt(30)*exp(2*I*ph)*cos(th)**2/(8*sqrt(pi))
+                                                   + sqrt(30)*exp(2*I*ph)/(8*sqrt(pi)))
+
+    assert diff(Ynm(n, m, th, ph), th) == (m*cot(th)*Ynm(n, m, th, ph)
+                                           + sqrt((-m + n)*(m + n + 1))*exp(-I*ph)*Ynm(n, m + 1, th, ph))
+    assert diff(Ynm(n, m, th, ph), ph) == I*m*Ynm(n, m, th, ph)
+
+    assert conjugate(Ynm(n, m, th, ph)) == (-1)**(2*m)*exp(-2*I*m*ph)*Ynm(n, m, th, ph)
+
+    assert Ynm(n, m, -th, ph) == Ynm(n, m, th, ph)
+    assert Ynm(n, m, th, -ph) == exp(-2*I*m*ph)*Ynm(n, m, th, ph)
+    assert Ynm(n, -m, th, ph) == (-1)**m*exp(-2*I*m*ph)*Ynm(n, m, th, ph)
+
+
+def test_Ynm_c():
+    th, ph = Symbol("theta", real=True), Symbol("phi", real=True)
+    from sympy.abc import n,m
+
+    assert Ynm_c(n, m, th, ph) == (-1)**(2*m)*exp(-2*I*m*ph)*Ynm(n, m, th, ph)
+
+
+def test_Znm():
+    # https://en.wikipedia.org/wiki/Solid_harmonics#List_of_lowest_functions
+    th, ph = Symbol("theta", real=True), Symbol("phi", real=True)
+
+    assert Znm(0, 0, th, ph) == Ynm(0, 0, th, ph)
+    assert Znm(1, -1, th, ph) == (-sqrt(2)*I*(Ynm(1, 1, th, ph)
+                                  - exp(-2*I*ph)*Ynm(1, 1, th, ph))/2)
+    assert Znm(1, 0, th, ph) == Ynm(1, 0, th, ph)
+    assert Znm(1, 1, th, ph) == (sqrt(2)*(Ynm(1, 1, th, ph)
+                                 + exp(-2*I*ph)*Ynm(1, 1, th, ph))/2)
+    assert Znm(0, 0, th, ph).expand(func=True) == 1/(2*sqrt(pi))
+    assert Znm(1, -1, th, ph).expand(func=True) == (sqrt(3)*I*sin(th)*exp(I*ph)/(4*sqrt(pi))
+                                                    - sqrt(3)*I*sin(th)*exp(-I*ph)/(4*sqrt(pi)))
+    assert Znm(1, 0, th, ph).expand(func=True) == sqrt(3)*cos(th)/(2*sqrt(pi))
+    assert Znm(1, 1, th, ph).expand(func=True) == (-sqrt(3)*sin(th)*exp(I*ph)/(4*sqrt(pi))
+                                                   - sqrt(3)*sin(th)*exp(-I*ph)/(4*sqrt(pi)))
+    assert Znm(2, -1, th, ph).expand(func=True) == (sqrt(15)*I*sin(th)*exp(I*ph)*cos(th)/(4*sqrt(pi))
+                                                    - sqrt(15)*I*sin(th)*exp(-I*ph)*cos(th)/(4*sqrt(pi)))
+    assert Znm(2, 0, th, ph).expand(func=True) == 3*sqrt(5)*cos(th)**2/(4*sqrt(pi)) - sqrt(5)/(4*sqrt(pi))
+    assert Znm(2, 1, th, ph).expand(func=True) == (-sqrt(15)*sin(th)*exp(I*ph)*cos(th)/(4*sqrt(pi))
+                                                   - sqrt(15)*sin(th)*exp(-I*ph)*cos(th)/(4*sqrt(pi)))

rl/Lib/site-packages/sympy/functions/special/tests/test_tensor_functions.py

@@ -0,0 +1,145 @@
+from sympy.core.relational import Ne
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.functions.elementary.complexes import (adjoint, conjugate, transpose)
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.special.tensor_functions import (Eijk, KroneckerDelta, LeviCivita)
+
+from sympy.physics.secondquant import evaluate_deltas, F
+
+x, y = symbols('x y')
+
+
+def test_levicivita():
+    assert Eijk(1, 2, 3) == LeviCivita(1, 2, 3)
+    assert LeviCivita(1, 2, 3) == 1
+    assert LeviCivita(int(1), int(2), int(3)) == 1
+    assert LeviCivita(1, 3, 2) == -1
+    assert LeviCivita(1, 2, 2) == 0
+    i, j, k = symbols('i j k')
+    assert LeviCivita(i, j, k) == LeviCivita(i, j, k, evaluate=False)
+    assert LeviCivita(i, j, i) == 0
+    assert LeviCivita(1, i, i) == 0
+    assert LeviCivita(i, j, k).doit() == (j - i)*(k - i)*(k - j)/2
+    assert LeviCivita(1, 2, 3, 1) == 0
+    assert LeviCivita(4, 5, 1, 2, 3) == 1
+    assert LeviCivita(4, 5, 2, 1, 3) == -1
+
+    assert LeviCivita(i, j, k).is_integer is True
+
+    assert adjoint(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
+    assert conjugate(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
+    assert transpose(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
+
+
+def test_kronecker_delta():
+    i, j = symbols('i j')
+    k = Symbol('k', nonzero=True)
+    assert KroneckerDelta(1, 1) == 1
+    assert KroneckerDelta(1, 2) == 0
+    assert KroneckerDelta(k, 0) == 0
+    assert KroneckerDelta(x, x) == 1
+    assert KroneckerDelta(x**2 - y**2, x**2 - y**2) == 1
+    assert KroneckerDelta(i, i) == 1
+    assert KroneckerDelta(i, i + 1) == 0
+    assert KroneckerDelta(0, 0) == 1
+    assert KroneckerDelta(0, 1) == 0
+    assert KroneckerDelta(i + k, i) == 0
+    assert KroneckerDelta(i + k, i + k) == 1
+    assert KroneckerDelta(i + k, i + 1 + k) == 0
+    assert KroneckerDelta(i, j).subs({"i": 1, "j": 0}) == 0
+    assert KroneckerDelta(i, j).subs({"i": 3, "j": 3}) == 1
+
+    assert KroneckerDelta(i, j)**0 == 1
+    for n in range(1, 10):
+        assert KroneckerDelta(i, j)**n == KroneckerDelta(i, j)
+        assert KroneckerDelta(i, j)**-n == 1/KroneckerDelta(i, j)
+
+    assert KroneckerDelta(i, j).is_integer is True
+
+    assert adjoint(KroneckerDelta(i, j)) == KroneckerDelta(i, j)
+    assert conjugate(KroneckerDelta(i, j)) == KroneckerDelta(i, j)
+    assert transpose(KroneckerDelta(i, j)) == KroneckerDelta(i, j)
+    # to test if canonical
+    assert (KroneckerDelta(i, j) == KroneckerDelta(j, i)) == True
+
+    assert KroneckerDelta(i, j).rewrite(Piecewise) == Piecewise((0, Ne(i, j)), (1, True))
+
+    # Tests with range:
+    assert KroneckerDelta(i, j, (0, i)).args == (i, j, (0, i))
+    assert KroneckerDelta(i, j, (-j, i)).delta_range == (-j, i)
+
+    # If index is out of range, return zero:
+    assert KroneckerDelta(i, j, (0, i-1)) == 0
+    assert KroneckerDelta(-1, j, (0, i-1)) == 0
+    assert KroneckerDelta(j, -1, (0, i-1)) == 0
+    assert KroneckerDelta(j, i, (0, i-1)) == 0
+
+
+def test_kronecker_delta_secondquant():
+    """secondquant-specific methods"""
+    D = KroneckerDelta
+    i, j, v, w = symbols('i j v w', below_fermi=True, cls=Dummy)
+    a, b, t, u = symbols('a b t u', above_fermi=True, cls=Dummy)
+    p, q, r, s = symbols('p q r s', cls=Dummy)
+
+    assert D(i, a) == 0
+    assert D(i, t) == 0
+
+    assert D(i, j).is_above_fermi is False
+    assert D(a, b).is_above_fermi is True
+    assert D(p, q).is_above_fermi is True
+    assert D(i, q).is_above_fermi is False
+    assert D(q, i).is_above_fermi is False
+    assert D(q, v).is_above_fermi is False
+    assert D(a, q).is_above_fermi is True
+
+    assert D(i, j).is_below_fermi is True
+    assert D(a, b).is_below_fermi is False
+    assert D(p, q).is_below_fermi is True
+    assert D(p, j).is_below_fermi is True
+    assert D(q, b).is_below_fermi is False
+
+    assert D(i, j).is_only_above_fermi is False
+    assert D(a, b).is_only_above_fermi is True
+    assert D(p, q).is_only_above_fermi is False
+    assert D(i, q).is_only_above_fermi is False
+    assert D(q, i).is_only_above_fermi is False
+    assert D(a, q).is_only_above_fermi is True
+
+    assert D(i, j).is_only_below_fermi is True
+    assert D(a, b).is_only_below_fermi is False
+    assert D(p, q).is_only_below_fermi is False
+    assert D(p, j).is_only_below_fermi is True
+    assert D(q, b).is_only_below_fermi is False
+
+    assert not D(i, q).indices_contain_equal_information
+    assert not D(a, q).indices_contain_equal_information
+    assert D(p, q).indices_contain_equal_information
+    assert D(a, b).indices_contain_equal_information
+    assert D(i, j).indices_contain_equal_information
+
+    assert D(q, b).preferred_index == b
+    assert D(q, b).killable_index == q
+    assert D(q, t).preferred_index == t
+    assert D(q, t).killable_index == q
+    assert D(q, i).preferred_index == i
+    assert D(q, i).killable_index == q
+    assert D(q, v).preferred_index == v
+    assert D(q, v).killable_index == q
+    assert D(q, p).preferred_index == p
+    assert D(q, p).killable_index == q
+
+    EV = evaluate_deltas
+    assert EV(D(a, q)*F(q)) == F(a)
+    assert EV(D(i, q)*F(q)) == F(i)
+    assert EV(D(a, q)*F(a)) == D(a, q)*F(a)
+    assert EV(D(i, q)*F(i)) == D(i, q)*F(i)
+    assert EV(D(a, b)*F(a)) == F(b)
+    assert EV(D(a, b)*F(b)) == F(a)
+    assert EV(D(i, j)*F(i)) == F(j)
+    assert EV(D(i, j)*F(j)) == F(i)
+    assert EV(D(p, q)*F(q)) == F(p)
+    assert EV(D(p, q)*F(p)) == F(q)
+    assert EV(D(p, j)*D(p, i)*F(i)) == F(j)
+    assert EV(D(p, j)*D(p, i)*F(j)) == F(i)
+    assert EV(D(p, q)*D(p, i))*F(i) == D(q, i)*F(i)

rl/Lib/site-packages/sympy/functions/special/tests/test_zeta_functions.py

@@ -0,0 +1,286 @@
+from sympy.concrete.summations import Sum
+from sympy.core.function import expand_func
+from sympy.core.numbers import (Float, I, Rational, nan, oo, pi, zoo)
+from sympy.core.singleton import S
+from sympy.core.symbol import Symbol
+from sympy.functions.elementary.complexes import (Abs, polar_lift)
+from sympy.functions.elementary.exponential import (exp, exp_polar, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.special.zeta_functions import (dirichlet_eta, lerchphi, polylog, riemann_xi, stieltjes, zeta)
+from sympy.series.order import O
+from sympy.core.function import ArgumentIndexError
+from sympy.functions.combinatorial.numbers import bernoulli, factorial, genocchi, harmonic
+from sympy.testing.pytest import raises
+from sympy.core.random import (test_derivative_numerically as td,
+                      random_complex_number as randcplx, verify_numerically)
+
+x = Symbol('x')
+a = Symbol('a')
+b = Symbol('b', negative=True)
+z = Symbol('z')
+s = Symbol('s')
+
+
+def test_zeta_eval():
+
+    assert zeta(nan) is nan
+    assert zeta(x, nan) is nan
+
+    assert zeta(0) == Rational(-1, 2)
+    assert zeta(0, x) == S.Half - x
+    assert zeta(0, b) == S.Half - b
+
+    assert zeta(1) is zoo
+    assert zeta(1, 2) is zoo
+    assert zeta(1, -7) is zoo
+    assert zeta(1, x) is zoo
+
+    assert zeta(2, 1) == pi**2/6
+    assert zeta(3, 1) == zeta(3)
+
+    assert zeta(2) == pi**2/6
+    assert zeta(4) == pi**4/90
+    assert zeta(6) == pi**6/945
+
+    assert zeta(4, 3) == pi**4/90 - Rational(17, 16)
+    assert zeta(7, 4) == zeta(7) - Rational(282251, 279936)
+    assert zeta(S.Half, 2).func == zeta
+    assert expand_func(zeta(S.Half, 2)) == zeta(S.Half) - 1
+    assert zeta(x, 3).func == zeta
+    assert expand_func(zeta(x, 3)) == zeta(x) - 1 - 1/2**x
+
+    assert zeta(2, 0) is nan
+    assert zeta(3, -1) is nan
+    assert zeta(4, -2) is nan
+
+    assert zeta(oo) == 1
+
+    assert zeta(-1) == Rational(-1, 12)
+    assert zeta(-2) == 0
+    assert zeta(-3) == Rational(1, 120)
+    assert zeta(-4) == 0
+    assert zeta(-5) == Rational(-1, 252)
+
+    assert zeta(-1, 3) == Rational(-37, 12)
+    assert zeta(-1, 7) == Rational(-253, 12)
+    assert zeta(-1, -4) == Rational(-121, 12)
+    assert zeta(-1, -9) == Rational(-541, 12)
+
+    assert zeta(-4, 3) == -17
+    assert zeta(-4, -8) == 8772
+
+    assert zeta(0, 1) == Rational(-1, 2)
+    assert zeta(0, -1) == Rational(3, 2)
+
+    assert zeta(0, 2) == Rational(-3, 2)
+    assert zeta(0, -2) == Rational(5, 2)
+
+    assert zeta(
+        3).evalf(20).epsilon_eq(Float("1.2020569031595942854", 20), 1e-19)
+
+
+def test_zeta_series():
+    assert zeta(x, a).series(a, z, 2) == \
+        zeta(x, z) - x*(a-z)*zeta(x+1, z) + O((a-z)**2, (a, z))
+
+
+def test_dirichlet_eta_eval():
+    assert dirichlet_eta(0) == S.Half
+    assert dirichlet_eta(-1) == Rational(1, 4)
+    assert dirichlet_eta(1) == log(2)
+    assert dirichlet_eta(1, S.Half).simplify() == pi/2
+    assert dirichlet_eta(1, 2) == 1 - log(2)
+    assert dirichlet_eta(2) == pi**2/12
+    assert dirichlet_eta(4) == pi**4*Rational(7, 720)
+    assert str(dirichlet_eta(I).evalf(n=10)) == '0.5325931818 + 0.2293848577*I'
+    assert str(dirichlet_eta(I, I).evalf(n=10)) == '3.462349253 + 0.220285771*I'
+
+
+def test_riemann_xi_eval():
+    assert riemann_xi(2) == pi/6
+    assert riemann_xi(0) == Rational(1, 2)
+    assert riemann_xi(1) == Rational(1, 2)
+    assert riemann_xi(3).rewrite(zeta) == 3*zeta(3)/(2*pi)
+    assert riemann_xi(4) == pi**2/15
+
+
+def test_rewriting():
+    from sympy.functions.elementary.piecewise import Piecewise
+    assert isinstance(dirichlet_eta(x).rewrite(zeta), Piecewise)
+    assert isinstance(dirichlet_eta(x).rewrite(genocchi), Piecewise)
+    assert zeta(x).rewrite(dirichlet_eta) == dirichlet_eta(x)/(1 - 2**(1 - x))
+    assert zeta(x).rewrite(dirichlet_eta, a=2) == zeta(x)
+    assert verify_numerically(dirichlet_eta(x), dirichlet_eta(x).rewrite(zeta), x)
+    assert verify_numerically(dirichlet_eta(x), dirichlet_eta(x).rewrite(genocchi), x)
+    assert verify_numerically(zeta(x), zeta(x).rewrite(dirichlet_eta), x)
+
+    assert zeta(x, a).rewrite(lerchphi) == lerchphi(1, x, a)
+    assert polylog(s, z).rewrite(lerchphi) == lerchphi(z, s, 1)*z
+
+    assert lerchphi(1, x, a).rewrite(zeta) == zeta(x, a)
+    assert z*lerchphi(z, s, 1).rewrite(polylog) == polylog(s, z)
+
+
+def test_derivatives():
+    from sympy.core.function import Derivative
+    assert zeta(x, a).diff(x) == Derivative(zeta(x, a), x)
+    assert zeta(x, a).diff(a) == -x*zeta(x + 1, a)
+    assert lerchphi(
+        z, s, a).diff(z) == (lerchphi(z, s - 1, a) - a*lerchphi(z, s, a))/z
+    assert lerchphi(z, s, a).diff(a) == -s*lerchphi(z, s + 1, a)
+    assert polylog(s, z).diff(z) == polylog(s - 1, z)/z
+
+    b = randcplx()
+    c = randcplx()
+    assert td(zeta(b, x), x)
+    assert td(polylog(b, z), z)
+    assert td(lerchphi(c, b, x), x)
+    assert td(lerchphi(x, b, c), x)
+    raises(ArgumentIndexError, lambda: lerchphi(c, b, x).fdiff(2))
+    raises(ArgumentIndexError, lambda: lerchphi(c, b, x).fdiff(4))
+    raises(ArgumentIndexError, lambda: polylog(b, z).fdiff(1))
+    raises(ArgumentIndexError, lambda: polylog(b, z).fdiff(3))
+
+
+def myexpand(func, target):
+    expanded = expand_func(func)
+    if target is not None:
+        return expanded == target
+    if expanded == func:  # it didn't expand
+        return False
+
+    # check to see that the expanded and original evaluate to the same value
+    subs = {}
+    for a in func.free_symbols:
+        subs[a] = randcplx()
+    return abs(func.subs(subs).n()
+               - expanded.replace(exp_polar, exp).subs(subs).n()) < 1e-10
+
+
+def test_polylog_expansion():
+    assert polylog(s, 0) == 0
+    assert polylog(s, 1) == zeta(s)
+    assert polylog(s, -1) == -dirichlet_eta(s)
+    assert polylog(s, exp_polar(I*pi*Rational(4, 3))) == polylog(s, exp(I*pi*Rational(4, 3)))
+    assert polylog(s, exp_polar(I*pi)/3) == polylog(s, exp(I*pi)/3)
+
+    assert myexpand(polylog(1, z), -log(1 - z))
+    assert myexpand(polylog(0, z), z/(1 - z))
+    assert myexpand(polylog(-1, z), z/(1 - z)**2)
+    assert ((1-z)**3 * expand_func(polylog(-2, z))).simplify() == z*(1 + z)
+    assert myexpand(polylog(-5, z), None)
+
+
+def test_polylog_series():
+    assert polylog(1, z).series(z, n=5) == z + z**2/2 + z**3/3 + z**4/4 + O(z**5)
+    assert polylog(1, sqrt(z)).series(z, n=3) == z/2 + z**2/4 + sqrt(z)\
+        + z**(S(3)/2)/3 + z**(S(5)/2)/5 + O(z**3)
+
+    # https://github.com/sympy/sympy/issues/9497
+    assert polylog(S(3)/2, -z).series(z, 0, 5) == -z + sqrt(2)*z**2/4\
+        - sqrt(3)*z**3/9 + z**4/8 + O(z**5)
+
+
+def test_issue_8404():
+    i = Symbol('i', integer=True)
+    assert Abs(Sum(1/(3*i + 1)**2, (i, 0, S.Infinity)).doit().n(4)
+        - 1.122) < 0.001
+
+
+def test_polylog_values():
+    assert polylog(2, 2) == pi**2/4 - I*pi*log(2)
+    assert polylog(2, S.Half) == pi**2/12 - log(2)**2/2
+    for z in [S.Half, 2, (sqrt(5)-1)/2, -(sqrt(5)-1)/2, -(sqrt(5)+1)/2, (3-sqrt(5))/2]:
+        assert Abs(polylog(2, z).evalf() - polylog(2, z, evaluate=False).evalf()) < 1e-15
+    z = Symbol("z")
+    for s in [-1, 0]:
+        for _ in range(10):
+            assert verify_numerically(polylog(s, z), polylog(s, z, evaluate=False),
+                                      z, a=-3, b=-2, c=S.Half, d=2)
+            assert verify_numerically(polylog(s, z), polylog(s, z, evaluate=False),
+                                      z, a=2, b=-2, c=5, d=2)
+
+    from sympy.integrals.integrals import Integral
+    assert polylog(0, Integral(1, (x, 0, 1))) == -S.Half
+
+
+def test_lerchphi_expansion():
+    assert myexpand(lerchphi(1, s, a), zeta(s, a))
+    assert myexpand(lerchphi(z, s, 1), polylog(s, z)/z)
+
+    # direct summation
+    assert myexpand(lerchphi(z, -1, a), a/(1 - z) + z/(1 - z)**2)
+    assert myexpand(lerchphi(z, -3, a), None)
+    # polylog reduction
+    assert myexpand(lerchphi(z, s, S.Half),
+                    2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z)
+                              - polylog(s, polar_lift(-1)*sqrt(z))/sqrt(z)))
+    assert myexpand(lerchphi(z, s, 2), -1/z + polylog(s, z)/z**2)
+    assert myexpand(lerchphi(z, s, Rational(3, 2)), None)
+    assert myexpand(lerchphi(z, s, Rational(7, 3)), None)
+    assert myexpand(lerchphi(z, s, Rational(-1, 3)), None)
+    assert myexpand(lerchphi(z, s, Rational(-5, 2)), None)
+
+    # hurwitz zeta reduction
+    assert myexpand(lerchphi(-1, s, a),
+                    2**(-s)*zeta(s, a/2) - 2**(-s)*zeta(s, (a + 1)/2))
+    assert myexpand(lerchphi(I, s, a), None)
+    assert myexpand(lerchphi(-I, s, a), None)
+    assert myexpand(lerchphi(exp(I*pi*Rational(2, 5)), s, a), None)
+
+
+def test_stieltjes():
+    assert isinstance(stieltjes(x), stieltjes)
+    assert isinstance(stieltjes(x, a), stieltjes)
+
+    # Zero'th constant EulerGamma
+    assert stieltjes(0) == S.EulerGamma
+    assert stieltjes(0, 1) == S.EulerGamma
+
+    # Not defined
+    assert stieltjes(nan) is nan
+    assert stieltjes(0, nan) is nan
+    assert stieltjes(-1) is S.ComplexInfinity
+    assert stieltjes(1.5) is S.ComplexInfinity
+    assert stieltjes(z, 0) is S.ComplexInfinity
+    assert stieltjes(z, -1) is S.ComplexInfinity
+
+
+def test_stieltjes_evalf():
+    assert abs(stieltjes(0).evalf() - 0.577215664) < 1E-9
+    assert abs(stieltjes(0, 0.5).evalf() - 1.963510026) < 1E-9
+    assert abs(stieltjes(1, 2).evalf() + 0.072815845) < 1E-9
+
+
+def test_issue_10475():
+    a = Symbol('a', extended_real=True)
+    b = Symbol('b', extended_positive=True)
+    s = Symbol('s', zero=False)
+
+    assert zeta(2 + I).is_finite
+    assert zeta(1).is_finite is False
+    assert zeta(x).is_finite is None
+    assert zeta(x + I).is_finite is None
+    assert zeta(a).is_finite is None
+    assert zeta(b).is_finite is None
+    assert zeta(-b).is_finite is True
+    assert zeta(b**2 - 2*b + 1).is_finite is None
+    assert zeta(a + I).is_finite is True
+    assert zeta(b + 1).is_finite is True
+    assert zeta(s + 1).is_finite is True
+
+
+def test_issue_14177():
+    n = Symbol('n', nonnegative=True, integer=True)
+
+    assert zeta(-n).rewrite(bernoulli) == bernoulli(n+1) / (-n-1)
+    assert zeta(-n, a).rewrite(bernoulli) == bernoulli(n+1, a) / (-n-1)
+    z2n = -(2*I*pi)**(2*n)*bernoulli(2*n) / (2*factorial(2*n))
+    assert zeta(2*n).rewrite(bernoulli) == z2n
+    assert expand_func(zeta(s, n+1)) == zeta(s) - harmonic(n, s)
+    assert expand_func(zeta(-b, -n)) is nan
+    assert expand_func(zeta(-b, n)) == zeta(-b, n)
+
+    n = Symbol('n')
+
+    assert zeta(2*n) == zeta(2*n) # As sign of z (= 2*n) is not determined

rl/Lib/site-packages/sympy/functions/special/zeta_functions.py

@@ -0,0 +1,787 @@
+""" Riemann zeta and related function. """
+
+from sympy.core.add import Add
+from sympy.core.cache import cacheit
+from sympy.core.function import ArgumentIndexError, expand_mul, Function
+from sympy.core.numbers import pi, I, Integer
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.core.symbol import Dummy
+from sympy.core.sympify import sympify
+from sympy.functions.combinatorial.numbers import bernoulli, factorial, genocchi, harmonic
+from sympy.functions.elementary.complexes import re, unpolarify, Abs, polar_lift
+from sympy.functions.elementary.exponential import log, exp_polar, exp
+from sympy.functions.elementary.integers import ceiling, floor
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.polys.polytools import Poly
+
+###############################################################################
+###################### LERCH TRANSCENDENT #####################################
+###############################################################################
+
+
+class lerchphi(Function):
+    r"""
+    Lerch transcendent (Lerch phi function).
+
+    Explanation
+    ===========
+
+    For $\operatorname{Re}(a) > 0$, $|z| < 1$ and $s \in \mathbb{C}$, the
+    Lerch transcendent is defined as
+
+    .. math :: \Phi(z, s, a) = \sum_{n=0}^\infty \frac{z^n}{(n + a)^s},
+
+    where the standard branch of the argument is used for $n + a$,
+    and by analytic continuation for other values of the parameters.
+
+    A commonly used related function is the Lerch zeta function, defined by
+
+    .. math:: L(q, s, a) = \Phi(e^{2\pi i q}, s, a).
+
+    **Analytic Continuation and Branching Behavior**
+
+    It can be shown that
+
+    .. math:: \Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.
+
+    This provides the analytic continuation to $\operatorname{Re}(a) \le 0$.
+
+    Assume now $\operatorname{Re}(a) > 0$. The integral representation
+
+    .. math:: \Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}}
+                                \frac{\mathrm{d}t}{\Gamma(s)}
+
+    provides an analytic continuation to $\mathbb{C} - [1, \infty)$.
+    Finally, for $x \in (1, \infty)$ we find
+
+    .. math:: \lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a)
+             -\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a)
+             = \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},
+
+    using the standard branch for both $\log{x}$ and
+    $\log{\log{x}}$ (a branch of $\log{\log{x}}$ is needed to
+    evaluate $\log{x}^{s-1}$).
+    This concludes the analytic continuation. The Lerch transcendent is thus
+    branched at $z \in \{0, 1, \infty\}$ and
+    $a \in \mathbb{Z}_{\le 0}$. For fixed $z, a$ outside these
+    branch points, it is an entire function of $s$.
+
+    Examples
+    ========
+
+    The Lerch transcendent is a fairly general function, for this reason it does
+    not automatically evaluate to simpler functions. Use ``expand_func()`` to
+    achieve this.
+
+    If $z=1$, the Lerch transcendent reduces to the Hurwitz zeta function:
+
+    >>> from sympy import lerchphi, expand_func
+    >>> from sympy.abc import z, s, a
+    >>> expand_func(lerchphi(1, s, a))
+    zeta(s, a)
+
+    More generally, if $z$ is a root of unity, the Lerch transcendent
+    reduces to a sum of Hurwitz zeta functions:
+
+    >>> expand_func(lerchphi(-1, s, a))
+    zeta(s, a/2)/2**s - zeta(s, a/2 + 1/2)/2**s
+
+    If $a=1$, the Lerch transcendent reduces to the polylogarithm:
+
+    >>> expand_func(lerchphi(z, s, 1))
+    polylog(s, z)/z
+
+    More generally, if $a$ is rational, the Lerch transcendent reduces
+    to a sum of polylogarithms:
+
+    >>> from sympy import S
+    >>> expand_func(lerchphi(z, s, S(1)/2))
+    2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -
+                polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))
+    >>> expand_func(lerchphi(z, s, S(3)/2))
+    -2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -
+                          polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z
+
+    The derivatives with respect to $z$ and $a$ can be computed in
+    closed form:
+
+    >>> lerchphi(z, s, a).diff(z)
+    (-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z
+    >>> lerchphi(z, s, a).diff(a)
+    -s*lerchphi(z, s + 1, a)
+
+    See Also
+    ========
+
+    polylog, zeta
+
+    References
+    ==========
+
+    .. [1] Bateman, H.; Erdelyi, A. (1953), Higher Transcendental Functions,
+           Vol. I, New York: McGraw-Hill. Section 1.11.
+    .. [2] https://dlmf.nist.gov/25.14
+    .. [3] https://en.wikipedia.org/wiki/Lerch_transcendent
+
+    """
+
+    def _eval_expand_func(self, **hints):
+        z, s, a = self.args
+        if z == 1:
+            return zeta(s, a)
+        if s.is_Integer and s <= 0:
+            t = Dummy('t')
+            p = Poly((t + a)**(-s), t)
+            start = 1/(1 - t)
+            res = S.Zero
+            for c in reversed(p.all_coeffs()):
+                res += c*start
+                start = t*start.diff(t)
+            return res.subs(t, z)
+
+        if a.is_Rational:
+            # See section 18 of
+            #   Kelly B. Roach.  Hypergeometric Function Representations.
+            #   In: Proceedings of the 1997 International Symposium on Symbolic and
+            #   Algebraic Computation, pages 205-211, New York, 1997. ACM.
+            # TODO should something be polarified here?
+            add = S.Zero
+            mul = S.One
+            # First reduce a to the interaval (0, 1]
+            if a > 1:
+                n = floor(a)
+                if n == a:
+                    n -= 1
+                a -= n
+                mul = z**(-n)
+                add = Add(*[-z**(k - n)/(a + k)**s for k in range(n)])
+            elif a <= 0:
+                n = floor(-a) + 1
+                a += n
+                mul = z**n
+                add = Add(*[z**(n - 1 - k)/(a - k - 1)**s for k in range(n)])
+
+            m, n = S([a.p, a.q])
+            zet = exp_polar(2*pi*I/n)
+            root = z**(1/n)
+            up_zet = unpolarify(zet)
+            addargs = []
+            for k in range(n):
+                p = polylog(s, zet**k*root)
+                if isinstance(p, polylog):
+                    p = p._eval_expand_func(**hints)
+                addargs.append(p/(up_zet**k*root)**m)
+            return add + mul*n**(s - 1)*Add(*addargs)
+
+        # TODO use minpoly instead of ad-hoc methods when issue 5888 is fixed
+        if isinstance(z, exp) and (z.args[0]/(pi*I)).is_Rational or z in [-1, I, -I]:
+            # TODO reference?
+            if z == -1:
+                p, q = S([1, 2])
+            elif z == I:
+                p, q = S([1, 4])
+            elif z == -I:
+                p, q = S([-1, 4])
+            else:
+                arg = z.args[0]/(2*pi*I)
+                p, q = S([arg.p, arg.q])
+            return Add(*[exp(2*pi*I*k*p/q)/q**s*zeta(s, (k + a)/q)
+                         for k in range(q)])
+
+        return lerchphi(z, s, a)
+
+    def fdiff(self, argindex=1):
+        z, s, a = self.args
+        if argindex == 3:
+            return -s*lerchphi(z, s + 1, a)
+        elif argindex == 1:
+            return (lerchphi(z, s - 1, a) - a*lerchphi(z, s, a))/z
+        else:
+            raise ArgumentIndexError
+
+    def _eval_rewrite_helper(self, target):
+        res = self._eval_expand_func()
+        if res.has(target):
+            return res
+        else:
+            return self
+
+    def _eval_rewrite_as_zeta(self, z, s, a, **kwargs):
+        return self._eval_rewrite_helper(zeta)
+
+    def _eval_rewrite_as_polylog(self, z, s, a, **kwargs):
+        return self._eval_rewrite_helper(polylog)
+
+###############################################################################
+###################### POLYLOGARITHM ##########################################
+###############################################################################
+
+
+class polylog(Function):
+    r"""
+    Polylogarithm function.
+
+    Explanation
+    ===========
+
+    For $|z| < 1$ and $s \in \mathbb{C}$, the polylogarithm is
+    defined by
+
+    .. math:: \operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s},
+
+    where the standard branch of the argument is used for $n$. It admits
+    an analytic continuation which is branched at $z=1$ (notably not on the
+    sheet of initial definition), $z=0$ and $z=\infty$.
+
+    The name polylogarithm comes from the fact that for $s=1$, the
+    polylogarithm is related to the ordinary logarithm (see examples), and that
+
+    .. math:: \operatorname{Li}_{s+1}(z) =
+                    \int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.
+
+    The polylogarithm is a special case of the Lerch transcendent:
+
+    .. math:: \operatorname{Li}_{s}(z) = z \Phi(z, s, 1).
+
+    Examples
+    ========
+
+    For $z \in \{0, 1, -1\}$, the polylogarithm is automatically expressed
+    using other functions:
+
+    >>> from sympy import polylog
+    >>> from sympy.abc import s
+    >>> polylog(s, 0)
+    0
+    >>> polylog(s, 1)
+    zeta(s)
+    >>> polylog(s, -1)
+    -dirichlet_eta(s)
+
+    If $s$ is a negative integer, $0$ or $1$, the polylogarithm can be
+    expressed using elementary functions. This can be done using
+    ``expand_func()``:
+
+    >>> from sympy import expand_func
+    >>> from sympy.abc import z
+    >>> expand_func(polylog(1, z))
+    -log(1 - z)
+    >>> expand_func(polylog(0, z))
+    z/(1 - z)
+
+    The derivative with respect to $z$ can be computed in closed form:
+
+    >>> polylog(s, z).diff(z)
+    polylog(s - 1, z)/z
+
+    The polylogarithm can be expressed in terms of the lerch transcendent:
+
+    >>> from sympy import lerchphi
+    >>> polylog(s, z).rewrite(lerchphi)
+    z*lerchphi(z, s, 1)
+
+    See Also
+    ========
+
+    zeta, lerchphi
+
+    """
+
+    @classmethod
+    def eval(cls, s, z):
+        if z.is_number:
+            if z is S.One:
+                return zeta(s)
+            elif z is S.NegativeOne:
+                return -dirichlet_eta(s)
+            elif z is S.Zero:
+                return S.Zero
+            elif s == 2:
+                dilogtable = _dilogtable()
+                if z in dilogtable:
+                    return dilogtable[z]
+
+        if z.is_zero:
+            return S.Zero
+
+        # Make an effort to determine if z is 1 to avoid replacing into
+        # expression with singularity
+        zone = z.equals(S.One)
+
+        if zone:
+            return zeta(s)
+        elif zone is False:
+            # For s = 0 or -1 use explicit formulas to evaluate, but
+            # automatically expanding polylog(1, z) to -log(1-z) seems
+            # undesirable for summation methods based on hypergeometric
+            # functions
+            if s is S.Zero:
+                return z/(1 - z)
+            elif s is S.NegativeOne:
+                return z/(1 - z)**2
+            if s.is_zero:
+                return z/(1 - z)
+
+        # polylog is branched, but not over the unit disk
+        if z.has(exp_polar, polar_lift) and (zone or (Abs(z) <= S.One) == True):
+            return cls(s, unpolarify(z))
+
+    def fdiff(self, argindex=1):
+        s, z = self.args
+        if argindex == 2:
+            return polylog(s - 1, z)/z
+        raise ArgumentIndexError
+
+    def _eval_rewrite_as_lerchphi(self, s, z, **kwargs):
+        return z*lerchphi(z, s, 1)
+
+    def _eval_expand_func(self, **hints):
+        s, z = self.args
+        if s == 1:
+            return -log(1 - z)
+        if s.is_Integer and s <= 0:
+            u = Dummy('u')
+            start = u/(1 - u)
+            for _ in range(-s):
+                start = u*start.diff(u)
+            return expand_mul(start).subs(u, z)
+        return polylog(s, z)
+
+    def _eval_is_zero(self):
+        z = self.args[1]
+        if z.is_zero:
+            return True
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        from sympy.series.order import Order
+        nu, z = self.args
+
+        z0 = z.subs(x, 0)
+        if z0 is S.NaN:
+            z0 = z.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
+
+        if z0.is_zero:
+            # In case of powers less than 1, number of terms need to be computed
+            # separately to avoid repeated callings of _eval_nseries with wrong n
+            try:
+                _, exp = z.leadterm(x)
+            except (ValueError, NotImplementedError):
+                return self
+
+            if exp.is_positive:
+                newn = ceiling(n/exp)
+                o = Order(x**n, x)
+                r = z._eval_nseries(x, n, logx, cdir).removeO()
+                if r is S.Zero:
+                    return o
+
+                term = r
+                s = [term]
+                for k in range(2, newn):
+                    term *= r
+                    s.append(term/k**nu)
+                return Add(*s) + o
+
+        return super(polylog, self)._eval_nseries(x, n, logx, cdir)
+
+###############################################################################
+###################### HURWITZ GENERALIZED ZETA FUNCTION ######################
+###############################################################################
+
+
+class zeta(Function):
+    r"""
+    Hurwitz zeta function (or Riemann zeta function).
+
+    Explanation
+    ===========
+
+    For $\operatorname{Re}(a) > 0$ and $\operatorname{Re}(s) > 1$, this
+    function is defined as
+
+    .. math:: \zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s},
+
+    where the standard choice of argument for $n + a$ is used. For fixed
+    $a$ not a nonpositive integer the Hurwitz zeta function admits a
+    meromorphic continuation to all of $\mathbb{C}$; it is an unbranched
+    function with a simple pole at $s = 1$.
+
+    The Hurwitz zeta function is a special case of the Lerch transcendent:
+
+    .. math:: \zeta(s, a) = \Phi(1, s, a).
+
+    This formula defines an analytic continuation for all possible values of
+    $s$ and $a$ (also $\operatorname{Re}(a) < 0$), see the documentation of
+    :class:`lerchphi` for a description of the branching behavior.
+
+    If no value is passed for $a$ a default value of $a = 1$ is assumed,
+    yielding the Riemann zeta function.
+
+    Examples
+    ========
+
+    For $a = 1$ the Hurwitz zeta function reduces to the famous Riemann
+    zeta function:
+
+    .. math:: \zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.
+
+    >>> from sympy import zeta
+    >>> from sympy.abc import s
+    >>> zeta(s, 1)
+    zeta(s)
+    >>> zeta(s)
+    zeta(s)
+
+    The Riemann zeta function can also be expressed using the Dirichlet eta
+    function:
+
+    >>> from sympy import dirichlet_eta
+    >>> zeta(s).rewrite(dirichlet_eta)
+    dirichlet_eta(s)/(1 - 2**(1 - s))
+
+    The Riemann zeta function at nonnegative even and negative integer
+    values is related to the Bernoulli numbers and polynomials:
+
+    >>> zeta(2)
+    pi**2/6
+    >>> zeta(4)
+    pi**4/90
+    >>> zeta(0)
+    -1/2
+    >>> zeta(-1)
+    -1/12
+    >>> zeta(-4)
+    0
+
+    The specific formulae are:
+
+    .. math:: \zeta(2n) = -\frac{(2\pi i)^{2n} B_{2n}}{2(2n)!}
+    .. math:: \zeta(-n,a) = -\frac{B_{n+1}(a)}{n+1}
+
+    No closed-form expressions are known at positive odd integers, but
+    numerical evaluation is possible:
+
+    >>> zeta(3).n()
+    1.20205690315959
+
+    The derivative of $\zeta(s, a)$ with respect to $a$ can be computed:
+
+    >>> from sympy.abc import a
+    >>> zeta(s, a).diff(a)
+    -s*zeta(s + 1, a)
+
+    However the derivative with respect to $s$ has no useful closed form
+    expression:
+
+    >>> zeta(s, a).diff(s)
+    Derivative(zeta(s, a), s)
+
+    The Hurwitz zeta function can be expressed in terms of the Lerch
+    transcendent, :class:`~.lerchphi`:
+
+    >>> from sympy import lerchphi
+    >>> zeta(s, a).rewrite(lerchphi)
+    lerchphi(1, s, a)
+
+    See Also
+    ========
+
+    dirichlet_eta, lerchphi, polylog
+
+    References
+    ==========
+
+    .. [1] https://dlmf.nist.gov/25.11
+    .. [2] https://en.wikipedia.org/wiki/Hurwitz_zeta_function
+
+    """
+
+    @classmethod
+    def eval(cls, s, a=None):
+        if a is S.One:
+            return cls(s)
+        elif s is S.NaN or a is S.NaN:
+            return S.NaN
+        elif s is S.One:
+            return S.ComplexInfinity
+        elif s is S.Infinity:
+            return S.One
+        elif a is S.Infinity:
+            return S.Zero
+
+        sint = s.is_Integer
+        if a is None:
+            a = S.One
+        if sint and s.is_nonpositive:
+            return bernoulli(1-s, a) / (s-1)
+        elif a is S.One:
+            if sint and s.is_even:
+                return -(2*pi*I)**s * bernoulli(s) / (2*factorial(s))
+        elif sint and a.is_Integer and a.is_positive:
+            return cls(s) - harmonic(a-1, s)
+        elif a.is_Integer and a.is_nonpositive and \
+                (s.is_integer is False or s.is_nonpositive is False):
+            return S.NaN
+
+    def _eval_rewrite_as_bernoulli(self, s, a=1, **kwargs):
+        if a == 1 and s.is_integer and s.is_nonnegative and s.is_even:
+            return -(2*pi*I)**s * bernoulli(s) / (2*factorial(s))
+        return bernoulli(1-s, a) / (s-1)
+
+    def _eval_rewrite_as_dirichlet_eta(self, s, a=1, **kwargs):
+        if a != 1:
+            return self
+        s = self.args[0]
+        return dirichlet_eta(s)/(1 - 2**(1 - s))
+
+    def _eval_rewrite_as_lerchphi(self, s, a=1, **kwargs):
+        return lerchphi(1, s, a)
+
+    def _eval_is_finite(self):
+        arg_is_one = (self.args[0] - 1).is_zero
+        if arg_is_one is not None:
+            return not arg_is_one
+
+    def _eval_expand_func(self, **hints):
+        s = self.args[0]
+        a = self.args[1] if len(self.args) > 1 else S.One
+        if a.is_integer:
+            if a.is_positive:
+                return zeta(s) - harmonic(a-1, s)
+            if a.is_nonpositive and (s.is_integer is False or
+                    s.is_nonpositive is False):
+                return S.NaN
+        return self
+
+    def fdiff(self, argindex=1):
+        if len(self.args) == 2:
+            s, a = self.args
+        else:
+            s, a = self.args + (1,)
+        if argindex == 2:
+            return -s*zeta(s + 1, a)
+        else:
+            raise ArgumentIndexError
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        if len(self.args) == 2:
+            s, a = self.args
+        else:
+            s, a = self.args + (S.One,)
+
+        try:
+            c, e = a.leadterm(x)
+        except NotImplementedError:
+            return self
+
+        if e.is_negative and not s.is_positive:
+            raise NotImplementedError
+
+        return super(zeta, self)._eval_as_leading_term(x, logx, cdir)
+
+
+class dirichlet_eta(Function):
+    r"""
+    Dirichlet eta function.
+
+    Explanation
+    ===========
+
+    For $\operatorname{Re}(s) > 0$ and $0 < x \le 1$, this function is defined as
+
+    .. math:: \eta(s, a) = \sum_{n=0}^\infty \frac{(-1)^n}{(n+a)^s}.
+
+    It admits a unique analytic continuation to all of $\mathbb{C}$ for any
+    fixed $a$ not a nonpositive integer. It is an entire, unbranched function.
+
+    It can be expressed using the Hurwitz zeta function as
+
+    .. math:: \eta(s, a) = \zeta(s,a) - 2^{1-s} \zeta\left(s, \frac{a+1}{2}\right)
+
+    and using the generalized Genocchi function as
+
+    .. math:: \eta(s, a) = \frac{G(1-s, a)}{2(s-1)}.
+
+    In both cases the limiting value of $\log2 - \psi(a) + \psi\left(\frac{a+1}{2}\right)$
+    is used when $s = 1$.
+
+    Examples
+    ========
+
+    >>> from sympy import dirichlet_eta, zeta
+    >>> from sympy.abc import s
+    >>> dirichlet_eta(s).rewrite(zeta)
+    Piecewise((log(2), Eq(s, 1)), ((1 - 2**(1 - s))*zeta(s), True))
+
+    See Also
+    ========
+
+    zeta
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Dirichlet_eta_function
+    .. [2] Peter Luschny, "An introduction to the Bernoulli function",
+           https://arxiv.org/abs/2009.06743
+
+    """
+
+    @classmethod
+    def eval(cls, s, a=None):
+        if a is S.One:
+            return cls(s)
+        if a is None:
+            if s == 1:
+                return log(2)
+            z = zeta(s)
+            if not z.has(zeta):
+                return (1 - 2**(1-s)) * z
+            return
+        elif s == 1:
+            from sympy.functions.special.gamma_functions import digamma
+            return log(2) - digamma(a) + digamma((a+1)/2)
+        z1 = zeta(s, a)
+        z2 = zeta(s, (a+1)/2)
+        if not z1.has(zeta) and not z2.has(zeta):
+            return z1 - 2**(1-s) * z2
+
+    def _eval_rewrite_as_zeta(self, s, a=1, **kwargs):
+        from sympy.functions.special.gamma_functions import digamma
+        if a == 1:
+            return Piecewise((log(2), Eq(s, 1)), ((1 - 2**(1-s)) * zeta(s), True))
+        return Piecewise((log(2) - digamma(a) + digamma((a+1)/2), Eq(s, 1)),
+                (zeta(s, a) - 2**(1-s) * zeta(s, (a+1)/2), True))
+
+    def _eval_rewrite_as_genocchi(self, s, a=S.One, **kwargs):
+        from sympy.functions.special.gamma_functions import digamma
+        return Piecewise((log(2) - digamma(a) + digamma((a+1)/2), Eq(s, 1)),
+                (genocchi(1-s, a) / (2 * (s-1)), True))
+
+    def _eval_evalf(self, prec):
+        if all(i.is_number for i in self.args):
+            return self.rewrite(zeta)._eval_evalf(prec)
+
+
+class riemann_xi(Function):
+    r"""
+    Riemann Xi function.
+
+    Examples
+    ========
+
+    The Riemann Xi function is closely related to the Riemann zeta function.
+    The zeros of Riemann Xi function are precisely the non-trivial zeros
+    of the zeta function.
+
+    >>> from sympy import riemann_xi, zeta
+    >>> from sympy.abc import s
+    >>> riemann_xi(s).rewrite(zeta)
+    s*(s - 1)*gamma(s/2)*zeta(s)/(2*pi**(s/2))
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Riemann_Xi_function
+
+    """
+
+
+    @classmethod
+    def eval(cls, s):
+        from sympy.functions.special.gamma_functions import gamma
+        z = zeta(s)
+        if s in (S.Zero, S.One):
+            return S.Half
+
+        if not isinstance(z, zeta):
+            return s*(s - 1)*gamma(s/2)*z/(2*pi**(s/2))
+
+    def _eval_rewrite_as_zeta(self, s, **kwargs):
+        from sympy.functions.special.gamma_functions import gamma
+        return s*(s - 1)*gamma(s/2)*zeta(s)/(2*pi**(s/2))
+
+
+class stieltjes(Function):
+    r"""
+    Represents Stieltjes constants, $\gamma_{k}$ that occur in
+    Laurent Series expansion of the Riemann zeta function.
+
+    Examples
+    ========
+
+    >>> from sympy import stieltjes
+    >>> from sympy.abc import n, m
+    >>> stieltjes(n)
+    stieltjes(n)
+
+    The zero'th stieltjes constant:
+
+    >>> stieltjes(0)
+    EulerGamma
+    >>> stieltjes(0, 1)
+    EulerGamma
+
+    For generalized stieltjes constants:
+
+    >>> stieltjes(n, m)
+    stieltjes(n, m)
+
+    Constants are only defined for integers >= 0:
+
+    >>> stieltjes(-1)
+    zoo
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Stieltjes_constants
+
+    """
+
+    @classmethod
+    def eval(cls, n, a=None):
+        if a is not None:
+            a = sympify(a)
+            if a is S.NaN:
+                return S.NaN
+            if a.is_Integer and a.is_nonpositive:
+                return S.ComplexInfinity
+
+        if n.is_Number:
+            if n is S.NaN:
+                return S.NaN
+            elif n < 0:
+                return S.ComplexInfinity
+            elif not n.is_Integer:
+                return S.ComplexInfinity
+            elif n is S.Zero and a in [None, 1]:
+                return S.EulerGamma
+
+        if n.is_extended_negative:
+            return S.ComplexInfinity
+
+        if n.is_zero and a in [None, 1]:
+            return S.EulerGamma
+
+        if n.is_integer == False:
+            return S.ComplexInfinity
+
+
+@cacheit
+def _dilogtable():
+    return {
+        S.Half: pi**2/12 - log(2)**2/2,
+        Integer(2) : pi**2/4 - I*pi*log(2),
+        -(sqrt(5) - 1)/2 : -pi**2/15 + log((sqrt(5)-1)/2)**2/2,
+        -(sqrt(5) + 1)/2 : -pi**2/10 - log((sqrt(5)+1)/2)**2,
+        (3 - sqrt(5))/2 : pi**2/15 - log((sqrt(5)-1)/2)**2,
+        (sqrt(5) - 1)/2 : pi**2/10 - log((sqrt(5)-1)/2)**2,
+        I : I*S.Catalan - pi**2/48,
+        -I : -I*S.Catalan - pi**2/48,
+        1 - I : pi**2/16 - I*S.Catalan - pi*I/4*log(2),
+        1 + I : pi**2/16 + I*S.Catalan + pi*I/4*log(2),
+        (1 - I)/2 : -log(2)**2/8 + pi*I*log(2)/8 + 5*pi**2/96 - I*S.Catalan
+    }