""" This module contains various functions that are special cases
    of incomplete gamma functions. It should probably be renamed. """

from sympy.core import Add, S, sympify, cacheit, pi, I, Rational, EulerGamma
from sympy.core.function import Function, ArgumentIndexError, expand_mul
from sympy.core.relational import is_eq
from sympy.core.power import Pow
from sympy.core.symbol import Symbol
from sympy.functions.combinatorial.factorials import factorial, factorial2, RisingFactorial
from sympy.functions.elementary.complexes import  polar_lift, re, unpolarify
from sympy.functions.elementary.integers import ceiling, floor
from sympy.functions.elementary.miscellaneous import sqrt, root
from sympy.functions.elementary.exponential import exp, log, exp_polar
from sympy.functions.elementary.hyperbolic import cosh, sinh
from sympy.functions.elementary.trigonometric import cos, sin, sinc
from sympy.functions.special.hyper import hyper, meijerg

# TODO series expansions
# TODO see the "Note:" in Ei

# Helper function
def real_to_real_as_real_imag(self, deep=True, **hints):
    if self.args[0].is_extended_real:
        if deep:
            hints['complex'] = False
            return (self.expand(deep, **hints), S.Zero)
        else:
            return (self, S.Zero)
    if deep:
        x, y = self.args[0].expand(deep, **hints).as_real_imag()
    else:
        x, y = self.args[0].as_real_imag()
    re = (self.func(x + I*y) + self.func(x - I*y))/2
    im = (self.func(x + I*y) - self.func(x - I*y))/(2*I)
    return (re, im)


###############################################################################
################################ ERROR FUNCTION ###############################
###############################################################################


class erf(Function):
    r"""
    The Gauss error function.

    Explanation
    ===========

    This function is defined as:

    .. math ::
        \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \mathrm{d}t.

    Examples
    ========

    >>> from sympy import I, oo, erf
    >>> from sympy.abc import z

    Several special values are known:

    >>> erf(0)
    0
    >>> erf(oo)
    1
    >>> erf(-oo)
    -1
    >>> erf(I*oo)
    oo*I
    >>> erf(-I*oo)
    -oo*I

    In general one can pull out factors of -1 and $I$ from the argument:

    >>> erf(-z)
    -erf(z)

    The error function obeys the mirror symmetry:

    >>> from sympy import conjugate
    >>> conjugate(erf(z))
    erf(conjugate(z))

    Differentiation with respect to $z$ is supported:

    >>> from sympy import diff
    >>> diff(erf(z), z)
    2*exp(-z**2)/sqrt(pi)

    We can numerically evaluate the error function to arbitrary precision
    on the whole complex plane:

    >>> erf(4).evalf(30)
    0.999999984582742099719981147840

    >>> erf(-4*I).evalf(30)
    -1296959.73071763923152794095062*I

    See Also
    ========

    erfc: Complementary error function.
    erfi: Imaginary error function.
    erf2: Two-argument error function.
    erfinv: Inverse error function.
    erfcinv: Inverse Complementary error function.
    erf2inv: Inverse two-argument error function.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Error_function
    .. [2] http://dlmf.nist.gov/7
    .. [3] http://mathworld.wolfram.com/Erf.html
    .. [4] http://functions.wolfram.com/GammaBetaErf/Erf

    """

    unbranched = True

    def fdiff(self, argindex=1):
        if argindex == 1:
            return 2*exp(-self.args[0]**2)/sqrt(S.Pi)
        else:
            raise ArgumentIndexError(self, argindex)


    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.

        """
        return erfinv

    @classmethod
    def eval(cls, arg):
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.One
            elif arg is S.NegativeInfinity:
                return S.NegativeOne
            elif arg.is_zero:
                return S.Zero

        if isinstance(arg, erfinv):
             return arg.args[0]

        if isinstance(arg, erfcinv):
            return S.One - arg.args[0]

        if arg.is_zero:
            return S.Zero

        # Only happens with unevaluated erf2inv
        if isinstance(arg, erf2inv) and arg.args[0].is_zero:
            return arg.args[1]

        # Try to pull out factors of I
        t = arg.extract_multiplicatively(S.ImaginaryUnit)
        if t in (S.Infinity, S.NegativeInfinity):
            return arg

        # Try to pull out factors of -1
        if arg.could_extract_minus_sign():
            return -cls(-arg)

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            k = floor((n - 1)/S(2))
            if len(previous_terms) > 2:
                return -previous_terms[-2] * x**2 * (n - 2)/(n*k)
            else:
                return 2*S.NegativeOne**k * x**n/(n*factorial(k)*sqrt(S.Pi))

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def _eval_is_real(self):
        return self.args[0].is_extended_real

    def _eval_is_finite(self):
        if self.args[0].is_finite:
            return True
        else:
            return self.args[0].is_extended_real

    def _eval_is_zero(self):
        return self.args[0].is_zero

    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
        from sympy.functions.special.gamma_functions import uppergamma
        return sqrt(z**2)/z*(S.One - uppergamma(S.Half, z**2)/sqrt(S.Pi))

    def _eval_rewrite_as_fresnels(self, z, **kwargs):
        arg = (S.One - S.ImaginaryUnit)*z/sqrt(pi)
        return (S.One + S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))

    def _eval_rewrite_as_fresnelc(self, z, **kwargs):
        arg = (S.One - S.ImaginaryUnit)*z/sqrt(pi)
        return (S.One + S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))

    def _eval_rewrite_as_meijerg(self, z, **kwargs):
        return z/sqrt(pi)*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)

    def _eval_rewrite_as_hyper(self, z, **kwargs):
        return 2*z/sqrt(pi)*hyper([S.Half], [3*S.Half], -z**2)

    def _eval_rewrite_as_expint(self, z, **kwargs):
        return sqrt(z**2)/z - z*expint(S.Half, z**2)/sqrt(S.Pi)

    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
        from sympy.series.limits import limit
        if limitvar:
            lim = limit(z, limitvar, S.Infinity)
            if lim is S.NegativeInfinity:
                return S.NegativeOne + _erfs(-z)*exp(-z**2)
        return S.One - _erfs(z)*exp(-z**2)

    def _eval_rewrite_as_erfc(self, z, **kwargs):
        return S.One - erfc(z)

    def _eval_rewrite_as_erfi(self, z, **kwargs):
        return -I*erfi(I*z)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
        arg0 = arg.subs(x, 0)

        if arg0 is S.ComplexInfinity:
            arg0 = arg.limit(x, 0, dir='-' if cdir == -1 else '+')
        if x in arg.free_symbols and arg0.is_zero:
            return 2*arg/sqrt(pi)
        else:
            return self.func(arg0)

    def _eval_aseries(self, n, args0, x, logx):
        from sympy.series.order import Order
        point = args0[0]

        if point in [S.Infinity, S.NegativeInfinity]:
            z = self.args[0]

            try:
                _, ex = z.leadterm(x)
            except (ValueError, NotImplementedError):
                return self

            ex = -ex # as x->1/x for aseries
            if ex.is_positive:
                newn = ceiling(n/ex)
                s = [S.NegativeOne**k * factorial2(2*k - 1) / (z**(2*k + 1) * 2**k)
                     for k in range(0, newn)] + [Order(1/z**newn, x)]
                return S.One - (exp(-z**2)/sqrt(pi)) * Add(*s)

        return super(erf, self)._eval_aseries(n, args0, x, logx)

    as_real_imag = real_to_real_as_real_imag


class erfc(Function):
    r"""
    Complementary Error Function.

    Explanation
    ===========

    The function is defined as:

    .. math ::
        \mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \mathrm{d}t

    Examples
    ========

    >>> from sympy import I, oo, erfc
    >>> from sympy.abc import z

    Several special values are known:

    >>> erfc(0)
    1
    >>> erfc(oo)
    0
    >>> erfc(-oo)
    2
    >>> erfc(I*oo)
    -oo*I
    >>> erfc(-I*oo)
    oo*I

    The error function obeys the mirror symmetry:

    >>> from sympy import conjugate
    >>> conjugate(erfc(z))
    erfc(conjugate(z))

    Differentiation with respect to $z$ is supported:

    >>> from sympy import diff
    >>> diff(erfc(z), z)
    -2*exp(-z**2)/sqrt(pi)

    It also follows

    >>> erfc(-z)
    2 - erfc(z)

    We can numerically evaluate the complementary error function to arbitrary
    precision on the whole complex plane:

    >>> erfc(4).evalf(30)
    0.0000000154172579002800188521596734869

    >>> erfc(4*I).evalf(30)
    1.0 - 1296959.73071763923152794095062*I

    See Also
    ========

    erf: Gaussian error function.
    erfi: Imaginary error function.
    erf2: Two-argument error function.
    erfinv: Inverse error function.
    erfcinv: Inverse Complementary error function.
    erf2inv: Inverse two-argument error function.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Error_function
    .. [2] http://dlmf.nist.gov/7
    .. [3] http://mathworld.wolfram.com/Erfc.html
    .. [4] http://functions.wolfram.com/GammaBetaErf/Erfc

    """

    unbranched = True

    def fdiff(self, argindex=1):
        if argindex == 1:
            return -2*exp(-self.args[0]**2)/sqrt(S.Pi)
        else:
            raise ArgumentIndexError(self, argindex)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.

        """
        return erfcinv

    @classmethod
    def eval(cls, arg):
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.Zero
            elif arg.is_zero:
                return S.One

        if isinstance(arg, erfinv):
            return S.One - arg.args[0]

        if isinstance(arg, erfcinv):
            return arg.args[0]

        if arg.is_zero:
            return S.One

        # Try to pull out factors of I
        t = arg.extract_multiplicatively(S.ImaginaryUnit)
        if t in (S.Infinity, S.NegativeInfinity):
            return -arg

        # Try to pull out factors of -1
        if arg.could_extract_minus_sign():
            return 2 - cls(-arg)

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n == 0:
            return S.One
        elif n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            k = floor((n - 1)/S(2))
            if len(previous_terms) > 2:
                return -previous_terms[-2] * x**2 * (n - 2)/(n*k)
            else:
                return -2*S.NegativeOne**k * x**n/(n*factorial(k)*sqrt(S.Pi))

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def _eval_is_real(self):
        return self.args[0].is_extended_real

    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
        return self.rewrite(erf).rewrite("tractable", deep=True, limitvar=limitvar)

    def _eval_rewrite_as_erf(self, z, **kwargs):
        return S.One - erf(z)

    def _eval_rewrite_as_erfi(self, z, **kwargs):
        return S.One + I*erfi(I*z)

    def _eval_rewrite_as_fresnels(self, z, **kwargs):
        arg = (S.One - S.ImaginaryUnit)*z/sqrt(pi)
        return S.One - (S.One + S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))

    def _eval_rewrite_as_fresnelc(self, z, **kwargs):
        arg = (S.One-S.ImaginaryUnit)*z/sqrt(pi)
        return S.One - (S.One + S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))

    def _eval_rewrite_as_meijerg(self, z, **kwargs):
        return S.One - z/sqrt(pi)*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)

    def _eval_rewrite_as_hyper(self, z, **kwargs):
        return S.One - 2*z/sqrt(pi)*hyper([S.Half], [3*S.Half], -z**2)

    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
        from sympy.functions.special.gamma_functions import uppergamma
        return S.One - sqrt(z**2)/z*(S.One - uppergamma(S.Half, z**2)/sqrt(S.Pi))

    def _eval_rewrite_as_expint(self, z, **kwargs):
        return S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi)

    def _eval_expand_func(self, **hints):
        return self.rewrite(erf)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
        arg0 = arg.subs(x, 0)

        if arg0 is S.ComplexInfinity:
            arg0 = arg.limit(x, 0, dir='-' if cdir == -1 else '+')
        if arg0.is_zero:
            return S.One
        else:
            return self.func(arg0)

    as_real_imag = real_to_real_as_real_imag

    def _eval_aseries(self, n, args0, x, logx):
        return S.One - erf(*self.args)._eval_aseries(n, args0, x, logx)


class erfi(Function):
    r"""
    Imaginary error function.

    Explanation
    ===========

    The function erfi is defined as:

    .. math ::
        \mathrm{erfi}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2} \mathrm{d}t

    Examples
    ========

    >>> from sympy import I, oo, erfi
    >>> from sympy.abc import z

    Several special values are known:

    >>> erfi(0)
    0
    >>> erfi(oo)
    oo
    >>> erfi(-oo)
    -oo
    >>> erfi(I*oo)
    I
    >>> erfi(-I*oo)
    -I

    In general one can pull out factors of -1 and $I$ from the argument:

    >>> erfi(-z)
    -erfi(z)

    >>> from sympy import conjugate
    >>> conjugate(erfi(z))
    erfi(conjugate(z))

    Differentiation with respect to $z$ is supported:

    >>> from sympy import diff
    >>> diff(erfi(z), z)
    2*exp(z**2)/sqrt(pi)

    We can numerically evaluate the imaginary error function to arbitrary
    precision on the whole complex plane:

    >>> erfi(2).evalf(30)
    18.5648024145755525987042919132

    >>> erfi(-2*I).evalf(30)
    -0.995322265018952734162069256367*I

    See Also
    ========

    erf: Gaussian error function.
    erfc: Complementary error function.
    erf2: Two-argument error function.
    erfinv: Inverse error function.
    erfcinv: Inverse Complementary error function.
    erf2inv: Inverse two-argument error function.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Error_function
    .. [2] http://mathworld.wolfram.com/Erfi.html
    .. [3] http://functions.wolfram.com/GammaBetaErf/Erfi

    """

    unbranched = True

    def fdiff(self, argindex=1):
        if argindex == 1:
            return 2*exp(self.args[0]**2)/sqrt(S.Pi)
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, z):
        if z.is_Number:
            if z is S.NaN:
                return S.NaN
            elif z.is_zero:
                return S.Zero
            elif z is S.Infinity:
                return S.Infinity

        if z.is_zero:
            return S.Zero

        # Try to pull out factors of -1
        if z.could_extract_minus_sign():
            return -cls(-z)

        # Try to pull out factors of I
        nz = z.extract_multiplicatively(I)
        if nz is not None:
            if nz is S.Infinity:
                return I
            if isinstance(nz, erfinv):
                return I*nz.args[0]
            if isinstance(nz, erfcinv):
                return I*(S.One - nz.args[0])
            # Only happens with unevaluated erf2inv
            if isinstance(nz, erf2inv) and nz.args[0].is_zero:
                return I*nz.args[1]

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            k = floor((n - 1)/S(2))
            if len(previous_terms) > 2:
                return previous_terms[-2] * x**2 * (n - 2)/(n*k)
            else:
                return 2 * x**n/(n*factorial(k)*sqrt(S.Pi))

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def _eval_is_extended_real(self):
        return self.args[0].is_extended_real

    def _eval_is_zero(self):
        return self.args[0].is_zero

    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
        return self.rewrite(erf).rewrite("tractable", deep=True, limitvar=limitvar)

    def _eval_rewrite_as_erf(self, z, **kwargs):
        return -I*erf(I*z)

    def _eval_rewrite_as_erfc(self, z, **kwargs):
        return I*erfc(I*z) - I

    def _eval_rewrite_as_fresnels(self, z, **kwargs):
        arg = (S.One + S.ImaginaryUnit)*z/sqrt(pi)
        return (S.One - S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))

    def _eval_rewrite_as_fresnelc(self, z, **kwargs):
        arg = (S.One + S.ImaginaryUnit)*z/sqrt(pi)
        return (S.One - S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))

    def _eval_rewrite_as_meijerg(self, z, **kwargs):
        return z/sqrt(pi)*meijerg([S.Half], [], [0], [Rational(-1, 2)], -z**2)

    def _eval_rewrite_as_hyper(self, z, **kwargs):
        return 2*z/sqrt(pi)*hyper([S.Half], [3*S.Half], z**2)

    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
        from sympy.functions.special.gamma_functions import uppergamma
        return sqrt(-z**2)/z*(uppergamma(S.Half, -z**2)/sqrt(S.Pi) - S.One)

    def _eval_rewrite_as_expint(self, z, **kwargs):
        return sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(S.Pi)

    def _eval_expand_func(self, **hints):
        return self.rewrite(erf)

    as_real_imag = real_to_real_as_real_imag

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
        arg0 = arg.subs(x, 0)

        if x in arg.free_symbols and arg0.is_zero:
            return 2*arg/sqrt(pi)
        elif arg0.is_finite:
            return self.func(arg0)
        return self.func(arg)

    def _eval_aseries(self, n, args0, x, logx):
        from sympy.series.order import Order
        point = args0[0]

        if point is S.Infinity:
            z = self.args[0]
            s = [factorial2(2*k - 1) / (2**k * z**(2*k + 1))
                    for k in range(0, n)] + [Order(1/z**n, x)]
            return -S.ImaginaryUnit + (exp(z**2)/sqrt(pi)) * Add(*s)

        return super(erfi, self)._eval_aseries(n, args0, x, logx)


class erf2(Function):
    r"""
    Two-argument error function.

    Explanation
    ===========

    This function is defined as:

    .. math ::
        \mathrm{erf2}(x, y) = \frac{2}{\sqrt{\pi}} \int_x^y e^{-t^2} \mathrm{d}t

    Examples
    ========

    >>> from sympy import oo, erf2
    >>> from sympy.abc import x, y

    Several special values are known:

    >>> erf2(0, 0)
    0
    >>> erf2(x, x)
    0
    >>> erf2(x, oo)
    1 - erf(x)
    >>> erf2(x, -oo)
    -erf(x) - 1
    >>> erf2(oo, y)
    erf(y) - 1
    >>> erf2(-oo, y)
    erf(y) + 1

    In general one can pull out factors of -1:

    >>> erf2(-x, -y)
    -erf2(x, y)

    The error function obeys the mirror symmetry:

    >>> from sympy import conjugate
    >>> conjugate(erf2(x, y))
    erf2(conjugate(x), conjugate(y))

    Differentiation with respect to $x$, $y$ is supported:

    >>> from sympy import diff
    >>> diff(erf2(x, y), x)
    -2*exp(-x**2)/sqrt(pi)
    >>> diff(erf2(x, y), y)
    2*exp(-y**2)/sqrt(pi)

    See Also
    ========

    erf: Gaussian error function.
    erfc: Complementary error function.
    erfi: Imaginary error function.
    erfinv: Inverse error function.
    erfcinv: Inverse Complementary error function.
    erf2inv: Inverse two-argument error function.

    References
    ==========

    .. [1] http://functions.wolfram.com/GammaBetaErf/Erf2/

    """


    def fdiff(self, argindex):
        x, y = self.args
        if argindex == 1:
            return -2*exp(-x**2)/sqrt(S.Pi)
        elif argindex == 2:
            return 2*exp(-y**2)/sqrt(S.Pi)
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, x, y):
        chk = (S.Infinity, S.NegativeInfinity, S.Zero)
        if x is S.NaN or y is S.NaN:
            return S.NaN
        elif x == y:
            return S.Zero
        elif x in chk or y in chk:
            return erf(y) - erf(x)

        if isinstance(y, erf2inv) and y.args[0] == x:
            return y.args[1]

        if x.is_zero or y.is_zero or x.is_extended_real and x.is_infinite or \
                y.is_extended_real and y.is_infinite:
            return erf(y) - erf(x)

        #Try to pull out -1 factor
        sign_x = x.could_extract_minus_sign()
        sign_y = y.could_extract_minus_sign()
        if (sign_x and sign_y):
            return -cls(-x, -y)
        elif (sign_x or sign_y):
            return erf(y)-erf(x)

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate(), self.args[1].conjugate())

    def _eval_is_extended_real(self):
        return self.args[0].is_extended_real and self.args[1].is_extended_real

    def _eval_rewrite_as_erf(self, x, y, **kwargs):
        return erf(y) - erf(x)

    def _eval_rewrite_as_erfc(self, x, y, **kwargs):
        return erfc(x) - erfc(y)

    def _eval_rewrite_as_erfi(self, x, y, **kwargs):
        return I*(erfi(I*x)-erfi(I*y))

    def _eval_rewrite_as_fresnels(self, x, y, **kwargs):
        return erf(y).rewrite(fresnels) - erf(x).rewrite(fresnels)

    def _eval_rewrite_as_fresnelc(self, x, y, **kwargs):
        return erf(y).rewrite(fresnelc) - erf(x).rewrite(fresnelc)

    def _eval_rewrite_as_meijerg(self, x, y, **kwargs):
        return erf(y).rewrite(meijerg) - erf(x).rewrite(meijerg)

    def _eval_rewrite_as_hyper(self, x, y, **kwargs):
        return erf(y).rewrite(hyper) - erf(x).rewrite(hyper)

    def _eval_rewrite_as_uppergamma(self, x, y, **kwargs):
        from sympy.functions.special.gamma_functions import uppergamma
        return (sqrt(y**2)/y*(S.One - uppergamma(S.Half, y**2)/sqrt(S.Pi)) -
            sqrt(x**2)/x*(S.One - uppergamma(S.Half, x**2)/sqrt(S.Pi)))

    def _eval_rewrite_as_expint(self, x, y, **kwargs):
        return erf(y).rewrite(expint) - erf(x).rewrite(expint)

    def _eval_expand_func(self, **hints):
        return self.rewrite(erf)

    def _eval_is_zero(self):
        return is_eq(*self.args)

class erfinv(Function):
    r"""
    Inverse Error Function. The erfinv function is defined as:

    .. math ::
        \mathrm{erf}(x) = y \quad \Rightarrow \quad \mathrm{erfinv}(y) = x

    Examples
    ========

    >>> from sympy import erfinv
    >>> from sympy.abc import x

    Several special values are known:

    >>> erfinv(0)
    0
    >>> erfinv(1)
    oo

    Differentiation with respect to $x$ is supported:

    >>> from sympy import diff
    >>> diff(erfinv(x), x)
    sqrt(pi)*exp(erfinv(x)**2)/2

    We can numerically evaluate the inverse error function to arbitrary
    precision on [-1, 1]:

    >>> erfinv(0.2).evalf(30)
    0.179143454621291692285822705344

    See Also
    ========

    erf: Gaussian error function.
    erfc: Complementary error function.
    erfi: Imaginary error function.
    erf2: Two-argument error function.
    erfcinv: Inverse Complementary error function.
    erf2inv: Inverse two-argument error function.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions
    .. [2] http://functions.wolfram.com/GammaBetaErf/InverseErf/

    """


    def fdiff(self, argindex =1):
        if argindex == 1:
            return sqrt(S.Pi)*exp(self.func(self.args[0])**2)*S.Half
        else :
            raise ArgumentIndexError(self, argindex)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.

        """
        return erf

    @classmethod
    def eval(cls, z):
        if z is S.NaN:
            return S.NaN
        elif z is S.NegativeOne:
            return S.NegativeInfinity
        elif z.is_zero:
            return S.Zero
        elif z is S.One:
            return S.Infinity

        if isinstance(z, erf) and z.args[0].is_extended_real:
            return z.args[0]

        if z.is_zero:
            return S.Zero

        # Try to pull out factors of -1
        nz = z.extract_multiplicatively(-1)
        if nz is not None and (isinstance(nz, erf) and (nz.args[0]).is_extended_real):
            return -nz.args[0]

    def _eval_rewrite_as_erfcinv(self, z, **kwargs):
       return erfcinv(1-z)

    def _eval_is_zero(self):
        return self.args[0].is_zero


class erfcinv (Function):
    r"""
    Inverse Complementary Error Function. The erfcinv function is defined as:

    .. math ::
        \mathrm{erfc}(x) = y \quad \Rightarrow \quad \mathrm{erfcinv}(y) = x

    Examples
    ========

    >>> from sympy import erfcinv
    >>> from sympy.abc import x

    Several special values are known:

    >>> erfcinv(1)
    0
    >>> erfcinv(0)
    oo

    Differentiation with respect to $x$ is supported:

    >>> from sympy import diff
    >>> diff(erfcinv(x), x)
    -sqrt(pi)*exp(erfcinv(x)**2)/2

    See Also
    ========

    erf: Gaussian error function.
    erfc: Complementary error function.
    erfi: Imaginary error function.
    erf2: Two-argument error function.
    erfinv: Inverse error function.
    erf2inv: Inverse two-argument error function.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions
    .. [2] http://functions.wolfram.com/GammaBetaErf/InverseErfc/

    """


    def fdiff(self, argindex =1):
        if argindex == 1:
            return -sqrt(S.Pi)*exp(self.func(self.args[0])**2)*S.Half
        else:
            raise ArgumentIndexError(self, argindex)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.

        """
        return erfc

    @classmethod
    def eval(cls, z):
        if z is S.NaN:
            return S.NaN
        elif z.is_zero:
            return S.Infinity
        elif z is S.One:
            return S.Zero
        elif z == 2:
            return S.NegativeInfinity

        if z.is_zero:
            return S.Infinity

    def _eval_rewrite_as_erfinv(self, z, **kwargs):
        return erfinv(1-z)

    def _eval_is_zero(self):
        return (self.args[0] - 1).is_zero

    def _eval_is_infinite(self):
        return self.args[0].is_zero


class erf2inv(Function):
    r"""
    Two-argument Inverse error function. The erf2inv function is defined as:

    .. math ::
        \mathrm{erf2}(x, w) = y \quad \Rightarrow \quad \mathrm{erf2inv}(x, y) = w

    Examples
    ========

    >>> from sympy import erf2inv, oo
    >>> from sympy.abc import x, y

    Several special values are known:

    >>> erf2inv(0, 0)
    0
    >>> erf2inv(1, 0)
    1
    >>> erf2inv(0, 1)
    oo
    >>> erf2inv(0, y)
    erfinv(y)
    >>> erf2inv(oo, y)
    erfcinv(-y)

    Differentiation with respect to $x$ and $y$ is supported:

    >>> from sympy import diff
    >>> diff(erf2inv(x, y), x)
    exp(-x**2 + erf2inv(x, y)**2)
    >>> diff(erf2inv(x, y), y)
    sqrt(pi)*exp(erf2inv(x, y)**2)/2

    See Also
    ========

    erf: Gaussian error function.
    erfc: Complementary error function.
    erfi: Imaginary error function.
    erf2: Two-argument error function.
    erfinv: Inverse error function.
    erfcinv: Inverse complementary error function.

    References
    ==========

    .. [1] http://functions.wolfram.com/GammaBetaErf/InverseErf2/

    """


    def fdiff(self, argindex):
        x, y = self.args
        if argindex == 1:
            return exp(self.func(x,y)**2-x**2)
        elif argindex == 2:
            return sqrt(S.Pi)*S.Half*exp(self.func(x,y)**2)
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, x, y):
        if x is S.NaN or y is S.NaN:
            return S.NaN
        elif x.is_zero and y.is_zero:
            return S.Zero
        elif x.is_zero and y is S.One:
            return S.Infinity
        elif x is S.One and y.is_zero:
            return S.One
        elif x.is_zero:
            return erfinv(y)
        elif x is S.Infinity:
            return erfcinv(-y)
        elif y.is_zero:
            return x
        elif y is S.Infinity:
            return erfinv(x)

        if x.is_zero:
            if y.is_zero:
                return S.Zero
            else:
                return erfinv(y)
        if y.is_zero:
            return x

    def _eval_is_zero(self):
        x, y = self.args
        if x.is_zero and y.is_zero:
            return True

###############################################################################
#################### EXPONENTIAL INTEGRALS ####################################
###############################################################################

class Ei(Function):
    r"""
    The classical exponential integral.

    Explanation
    ===========

    For use in SymPy, this function is defined as

    .. math:: \operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!}
                                     + \log(x) + \gamma,

    where $\gamma$ is the Euler-Mascheroni constant.

    If $x$ is a polar number, this defines an analytic function on the
    Riemann surface of the logarithm. Otherwise this defines an analytic
    function in the cut plane $\mathbb{C} \setminus (-\infty, 0]$.

    **Background**

    The name exponential integral comes from the following statement:

    .. math:: \operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t

    If the integral is interpreted as a Cauchy principal value, this statement
    holds for $x > 0$ and $\operatorname{Ei}(x)$ as defined above.

    Examples
    ========

    >>> from sympy import Ei, polar_lift, exp_polar, I, pi
    >>> from sympy.abc import x

    >>> Ei(-1)
    Ei(-1)

    This yields a real value:

    >>> Ei(-1).n(chop=True)
    -0.219383934395520

    On the other hand the analytic continuation is not real:

    >>> Ei(polar_lift(-1)).n(chop=True)
    -0.21938393439552 + 3.14159265358979*I

    The exponential integral has a logarithmic branch point at the origin:

    >>> Ei(x*exp_polar(2*I*pi))
    Ei(x) + 2*I*pi

    Differentiation is supported:

    >>> Ei(x).diff(x)
    exp(x)/x

    The exponential integral is related to many other special functions.
    For example:

    >>> from sympy import expint, Shi
    >>> Ei(x).rewrite(expint)
    -expint(1, x*exp_polar(I*pi)) - I*pi
    >>> Ei(x).rewrite(Shi)
    Chi(x) + Shi(x)

    See Also
    ========

    expint: Generalised exponential integral.
    E1: Special case of the generalised exponential integral.
    li: Logarithmic integral.
    Li: Offset logarithmic integral.
    Si: Sine integral.
    Ci: Cosine integral.
    Shi: Hyperbolic sine integral.
    Chi: Hyperbolic cosine integral.
    uppergamma: Upper incomplete gamma function.

    References
    ==========

    .. [1] http://dlmf.nist.gov/6.6
    .. [2] https://en.wikipedia.org/wiki/Exponential_integral
    .. [3] Abramowitz & Stegun, section 5: http://people.math.sfu.ca/~cbm/aands/page_228.htm

    """


    @classmethod
    def eval(cls, z):
        if z.is_zero:
            return S.NegativeInfinity
        elif z is S.Infinity:
            return S.Infinity
        elif z is S.NegativeInfinity:
            return S.Zero

        if z.is_zero:
            return S.NegativeInfinity

        nz, n = z.extract_branch_factor()
        if n:
            return Ei(nz) + 2*I*pi*n

    def fdiff(self, argindex=1):
        arg = unpolarify(self.args[0])
        if argindex == 1:
            return exp(arg)/arg
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_evalf(self, prec):
        if (self.args[0]/polar_lift(-1)).is_positive:
            return Function._eval_evalf(self, prec) + (I*pi)._eval_evalf(prec)
        return Function._eval_evalf(self, prec)

    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
        from sympy.functions.special.gamma_functions import uppergamma
        # XXX this does not currently work usefully because uppergamma
        #     immediately turns into expint
        return -uppergamma(0, polar_lift(-1)*z) - I*pi

    def _eval_rewrite_as_expint(self, z, **kwargs):
        return -expint(1, polar_lift(-1)*z) - I*pi

    def _eval_rewrite_as_li(self, z, **kwargs):
        if isinstance(z, log):
            return li(z.args[0])
        # TODO:
        # Actually it only holds that:
        #  Ei(z) = li(exp(z))
        # for -pi < imag(z) <= pi
        return li(exp(z))

    def _eval_rewrite_as_Si(self, z, **kwargs):
        if z.is_negative:
            return Shi(z) + Chi(z) - I*pi
        else:
            return Shi(z) + Chi(z)
    _eval_rewrite_as_Ci = _eval_rewrite_as_Si
    _eval_rewrite_as_Chi = _eval_rewrite_as_Si
    _eval_rewrite_as_Shi = _eval_rewrite_as_Si

    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
        return exp(z) * _eis(z)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        x0 = self.args[0].limit(x, 0)
        if x0.is_zero:
            f = self._eval_rewrite_as_Si(*self.args)
            return f._eval_as_leading_term(x, logx=logx, cdir=cdir)
        return super()._eval_as_leading_term(x, logx=logx, cdir=cdir)

    def _eval_nseries(self, x, n, logx, cdir=0):
        x0 = self.args[0].limit(x, 0)
        if x0.is_zero:
            f = self._eval_rewrite_as_Si(*self.args)
            return f._eval_nseries(x, n, logx)
        return super()._eval_nseries(x, n, logx)

    def _eval_aseries(self, n, args0, x, logx):
        from sympy.series.order import Order
        point = args0[0]

        if point is S.Infinity:
            z = self.args[0]
            s = [factorial(k) / (z)**k for k in range(0, n)] + \
                    [Order(1/z**n, x)]
            return (exp(z)/z) * Add(*s)

        return super(Ei, self)._eval_aseries(n, args0, x, logx)


class expint(Function):
    r"""
    Generalized exponential integral.

    Explanation
    ===========

    This function is defined as

    .. math:: \operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),

    where $\Gamma(1 - \nu, z)$ is the upper incomplete gamma function
    (``uppergamma``).

    Hence for $z$ with positive real part we have

    .. math:: \operatorname{E}_\nu(z)
              =   \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t,

    which explains the name.

    The representation as an incomplete gamma function provides an analytic
    continuation for $\operatorname{E}_\nu(z)$. If $\nu$ is a
    non-positive integer, the exponential integral is thus an unbranched
    function of $z$, otherwise there is a branch point at the origin.
    Refer to the incomplete gamma function documentation for details of the
    branching behavior.

    Examples
    ========

    >>> from sympy import expint, S
    >>> from sympy.abc import nu, z

    Differentiation is supported. Differentiation with respect to $z$ further
    explains the name: for integral orders, the exponential integral is an
    iterated integral of the exponential function.

    >>> expint(nu, z).diff(z)
    -expint(nu - 1, z)

    Differentiation with respect to $\nu$ has no classical expression:

    >>> expint(nu, z).diff(nu)
    -z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, 1 - nu), ()), z)

    At non-postive integer orders, the exponential integral reduces to the
    exponential function:

    >>> expint(0, z)
    exp(-z)/z
    >>> expint(-1, z)
    exp(-z)/z + exp(-z)/z**2

    At half-integers it reduces to error functions:

    >>> expint(S(1)/2, z)
    sqrt(pi)*erfc(sqrt(z))/sqrt(z)

    At positive integer orders it can be rewritten in terms of exponentials
    and ``expint(1, z)``. Use ``expand_func()`` to do this:

    >>> from sympy import expand_func
    >>> expand_func(expint(5, z))
    z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24

    The generalised exponential integral is essentially equivalent to the
    incomplete gamma function:

    >>> from sympy import uppergamma
    >>> expint(nu, z).rewrite(uppergamma)
    z**(nu - 1)*uppergamma(1 - nu, z)

    As such it is branched at the origin:

    >>> from sympy import exp_polar, pi, I
    >>> expint(4, z*exp_polar(2*pi*I))
    I*pi*z**3/3 + expint(4, z)
    >>> expint(nu, z*exp_polar(2*pi*I))
    z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(1 - nu) + expint(nu, z)

    See Also
    ========

    Ei: Another related function called exponential integral.
    E1: The classical case, returns expint(1, z).
    li: Logarithmic integral.
    Li: Offset logarithmic integral.
    Si: Sine integral.
    Ci: Cosine integral.
    Shi: Hyperbolic sine integral.
    Chi: Hyperbolic cosine integral.
    uppergamma

    References
    ==========

    .. [1] http://dlmf.nist.gov/8.19
    .. [2] http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/
    .. [3] https://en.wikipedia.org/wiki/Exponential_integral

    """


    @classmethod
    def eval(cls, nu, z):
        from sympy.functions.special.gamma_functions import (gamma, uppergamma)
        nu2 = unpolarify(nu)
        if nu != nu2:
            return expint(nu2, z)
        if nu.is_Integer and nu <= 0 or (not nu.is_Integer and (2*nu).is_Integer):
            return unpolarify(expand_mul(z**(nu - 1)*uppergamma(1 - nu, z)))

        # Extract branching information. This can be deduced from what is
        # explained in lowergamma.eval().
        z, n = z.extract_branch_factor()
        if n is S.Zero:
            return
        if nu.is_integer:
            if not nu > 0:
                return
            return expint(nu, z) \
                - 2*pi*I*n*S.NegativeOne**(nu - 1)/factorial(nu - 1)*unpolarify(z)**(nu - 1)
        else:
            return (exp(2*I*pi*nu*n) - 1)*z**(nu - 1)*gamma(1 - nu) + expint(nu, z)

    def fdiff(self, argindex):
        nu, z = self.args
        if argindex == 1:
            return -z**(nu - 1)*meijerg([], [1, 1], [0, 0, 1 - nu], [], z)
        elif argindex == 2:
            return -expint(nu - 1, z)
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_rewrite_as_uppergamma(self, nu, z, **kwargs):
        from sympy.functions.special.gamma_functions import uppergamma
        return z**(nu - 1)*uppergamma(1 - nu, z)

    def _eval_rewrite_as_Ei(self, nu, z, **kwargs):
        if nu == 1:
            return -Ei(z*exp_polar(-I*pi)) - I*pi
        elif nu.is_Integer and nu > 1:
            # DLMF, 8.19.7
            x = -unpolarify(z)
            return x**(nu - 1)/factorial(nu - 1)*E1(z).rewrite(Ei) + \
                exp(x)/factorial(nu - 1) * \
                Add(*[factorial(nu - k - 2)*x**k for k in range(nu - 1)])
        else:
            return self

    def _eval_expand_func(self, **hints):
        return self.rewrite(Ei).rewrite(expint, **hints)

    def _eval_rewrite_as_Si(self, nu, z, **kwargs):
        if nu != 1:
            return self
        return Shi(z) - Chi(z)
    _eval_rewrite_as_Ci = _eval_rewrite_as_Si
    _eval_rewrite_as_Chi = _eval_rewrite_as_Si
    _eval_rewrite_as_Shi = _eval_rewrite_as_Si

    def _eval_nseries(self, x, n, logx, cdir=0):
        if not self.args[0].has(x):
            nu = self.args[0]
            if nu == 1:
                f = self._eval_rewrite_as_Si(*self.args)
                return f._eval_nseries(x, n, logx)
            elif nu.is_Integer and nu > 1:
                f = self._eval_rewrite_as_Ei(*self.args)
                return f._eval_nseries(x, n, logx)
        return super()._eval_nseries(x, n, logx)

    def _eval_aseries(self, n, args0, x, logx):
        from sympy.series.order import Order
        point = args0[1]
        nu = self.args[0]

        if point is S.Infinity:
            z = self.args[1]
            s = [S.NegativeOne**k * RisingFactorial(nu, k) / z**k for k in range(0, n)] + [Order(1/z**n, x)]
            return (exp(-z)/z) * Add(*s)

        return super(expint, self)._eval_aseries(n, args0, x, logx)


def E1(z):
    """
    Classical case of the generalized exponential integral.

    Explanation
    ===========

    This is equivalent to ``expint(1, z)``.

    Examples
    ========

    >>> from sympy import E1
    >>> E1(0)
    expint(1, 0)

    >>> E1(5)
    expint(1, 5)

    See Also
    ========

    Ei: Exponential integral.
    expint: Generalised exponential integral.
    li: Logarithmic integral.
    Li: Offset logarithmic integral.
    Si: Sine integral.
    Ci: Cosine integral.
    Shi: Hyperbolic sine integral.
    Chi: Hyperbolic cosine integral.

    """
    return expint(1, z)


class li(Function):
    r"""
    The classical logarithmic integral.

    Explanation
    ===========

    For use in SymPy, this function is defined as

    .. math:: \operatorname{li}(x) = \int_0^x \frac{1}{\log(t)} \mathrm{d}t \,.

    Examples
    ========

    >>> from sympy import I, oo, li
    >>> from sympy.abc import z

    Several special values are known:

    >>> li(0)
    0
    >>> li(1)
    -oo
    >>> li(oo)
    oo

    Differentiation with respect to $z$ is supported:

    >>> from sympy import diff
    >>> diff(li(z), z)
    1/log(z)

    Defining the ``li`` function via an integral:
    >>> from sympy import integrate
    >>> integrate(li(z))
    z*li(z) - Ei(2*log(z))

    >>> integrate(li(z),z)
    z*li(z) - Ei(2*log(z))


    The logarithmic integral can also be defined in terms of ``Ei``:

    >>> from sympy import Ei
    >>> li(z).rewrite(Ei)
    Ei(log(z))
    >>> diff(li(z).rewrite(Ei), z)
    1/log(z)

    We can numerically evaluate the logarithmic integral to arbitrary precision
    on the whole complex plane (except the singular points):

    >>> li(2).evalf(30)
    1.04516378011749278484458888919

    >>> li(2*I).evalf(30)
    1.0652795784357498247001125598 + 3.08346052231061726610939702133*I

    We can even compute Soldner's constant by the help of mpmath:

    >>> from mpmath import findroot
    >>> findroot(li, 2)
    1.45136923488338

    Further transformations include rewriting ``li`` in terms of
    the trigonometric integrals ``Si``, ``Ci``, ``Shi`` and ``Chi``:

    >>> from sympy import Si, Ci, Shi, Chi
    >>> li(z).rewrite(Si)
    -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))
    >>> li(z).rewrite(Ci)
    -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))
    >>> li(z).rewrite(Shi)
    -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))
    >>> li(z).rewrite(Chi)
    -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))

    See Also
    ========

    Li: Offset logarithmic integral.
    Ei: Exponential integral.
    expint: Generalised exponential integral.
    E1: Special case of the generalised exponential integral.
    Si: Sine integral.
    Ci: Cosine integral.
    Shi: Hyperbolic sine integral.
    Chi: Hyperbolic cosine integral.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Logarithmic_integral
    .. [2] http://mathworld.wolfram.com/LogarithmicIntegral.html
    .. [3] http://dlmf.nist.gov/6
    .. [4] http://mathworld.wolfram.com/SoldnersConstant.html

    """


    @classmethod
    def eval(cls, z):
        if z.is_zero:
            return S.Zero
        elif z is S.One:
            return S.NegativeInfinity
        elif z is S.Infinity:
            return S.Infinity
        if z.is_zero:
            return S.Zero

    def fdiff(self, argindex=1):
        arg = self.args[0]
        if argindex == 1:
            return S.One / log(arg)
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_conjugate(self):
        z = self.args[0]
        # Exclude values on the branch cut (-oo, 0)
        if not z.is_extended_negative:
            return self.func(z.conjugate())

    def _eval_rewrite_as_Li(self, z, **kwargs):
        return Li(z) + li(2)

    def _eval_rewrite_as_Ei(self, z, **kwargs):
        return Ei(log(z))

    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
        from sympy.functions.special.gamma_functions import uppergamma
        return (-uppergamma(0, -log(z)) +
                S.Half*(log(log(z)) - log(S.One/log(z))) - log(-log(z)))

    def _eval_rewrite_as_Si(self, z, **kwargs):
        return (Ci(I*log(z)) - I*Si(I*log(z)) -
                S.Half*(log(S.One/log(z)) - log(log(z))) - log(I*log(z)))

    _eval_rewrite_as_Ci = _eval_rewrite_as_Si

    def _eval_rewrite_as_Shi(self, z, **kwargs):
        return (Chi(log(z)) - Shi(log(z)) - S.Half*(log(S.One/log(z)) - log(log(z))))

    _eval_rewrite_as_Chi = _eval_rewrite_as_Shi

    def _eval_rewrite_as_hyper(self, z, **kwargs):
        return (log(z)*hyper((1, 1), (2, 2), log(z)) +
                S.Half*(log(log(z)) - log(S.One/log(z))) + S.EulerGamma)

    def _eval_rewrite_as_meijerg(self, z, **kwargs):
        return (-log(-log(z)) - S.Half*(log(S.One/log(z)) - log(log(z)))
                - meijerg(((), (1,)), ((0, 0), ()), -log(z)))

    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
        return z * _eis(log(z))

    def _eval_nseries(self, x, n, logx, cdir=0):
        z = self.args[0]
        s = [(log(z))**k / (factorial(k) * k) for k in range(1, n)]
        return S.EulerGamma + log(log(z)) + Add(*s)

    def _eval_is_zero(self):
        z = self.args[0]
        if z.is_zero:
            return True

class Li(Function):
    r"""
    The offset logarithmic integral.

    Explanation
    ===========

    For use in SymPy, this function is defined as

    .. math:: \operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)

    Examples
    ========

    >>> from sympy import Li
    >>> from sympy.abc import z

    The following special value is known:

    >>> Li(2)
    0

    Differentiation with respect to $z$ is supported:

    >>> from sympy import diff
    >>> diff(Li(z), z)
    1/log(z)

    The shifted logarithmic integral can be written in terms of $li(z)$:

    >>> from sympy import li
    >>> Li(z).rewrite(li)
    li(z) - li(2)

    We can numerically evaluate the logarithmic integral to arbitrary precision
    on the whole complex plane (except the singular points):

    >>> Li(2).evalf(30)
    0

    >>> Li(4).evalf(30)
    1.92242131492155809316615998938

    See Also
    ========

    li: Logarithmic integral.
    Ei: Exponential integral.
    expint: Generalised exponential integral.
    E1: Special case of the generalised exponential integral.
    Si: Sine integral.
    Ci: Cosine integral.
    Shi: Hyperbolic sine integral.
    Chi: Hyperbolic cosine integral.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Logarithmic_integral
    .. [2] http://mathworld.wolfram.com/LogarithmicIntegral.html
    .. [3] http://dlmf.nist.gov/6

    """


    @classmethod
    def eval(cls, z):
        if z is S.Infinity:
            return S.Infinity
        elif z == S(2):
            return S.Zero

    def fdiff(self, argindex=1):
        arg = self.args[0]
        if argindex == 1:
            return S.One / log(arg)
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_evalf(self, prec):
        return self.rewrite(li).evalf(prec)

    def _eval_rewrite_as_li(self, z, **kwargs):
        return li(z) - li(2)

    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
        return self.rewrite(li).rewrite("tractable", deep=True)

    def _eval_nseries(self, x, n, logx, cdir=0):
        f = self._eval_rewrite_as_li(*self.args)
        return f._eval_nseries(x, n, logx)

###############################################################################
#################### TRIGONOMETRIC INTEGRALS ##################################
###############################################################################

class TrigonometricIntegral(Function):
    """ Base class for trigonometric integrals. """


    @classmethod
    def eval(cls, z):
        if z is S.Zero:
            return cls._atzero
        elif z is S.Infinity:
            return cls._atinf()
        elif z is S.NegativeInfinity:
            return cls._atneginf()

        if z.is_zero:
            return cls._atzero

        nz = z.extract_multiplicatively(polar_lift(I))
        if nz is None and cls._trigfunc(0) == 0:
            nz = z.extract_multiplicatively(I)
        if nz is not None:
            return cls._Ifactor(nz, 1)
        nz = z.extract_multiplicatively(polar_lift(-I))
        if nz is not None:
            return cls._Ifactor(nz, -1)

        nz = z.extract_multiplicatively(polar_lift(-1))
        if nz is None and cls._trigfunc(0) == 0:
            nz = z.extract_multiplicatively(-1)
        if nz is not None:
            return cls._minusfactor(nz)

        nz, n = z.extract_branch_factor()
        if n == 0 and nz == z:
            return
        return 2*pi*I*n*cls._trigfunc(0) + cls(nz)

    def fdiff(self, argindex=1):
        arg = unpolarify(self.args[0])
        if argindex == 1:
            return self._trigfunc(arg)/arg
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_rewrite_as_Ei(self, z, **kwargs):
        return self._eval_rewrite_as_expint(z).rewrite(Ei)

    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
        from sympy.functions.special.gamma_functions import uppergamma
        return self._eval_rewrite_as_expint(z).rewrite(uppergamma)

    def _eval_nseries(self, x, n, logx, cdir=0):
        # NOTE this is fairly inefficient
        n += 1
        if self.args[0].subs(x, 0) != 0:
            return super()._eval_nseries(x, n, logx)
        baseseries = self._trigfunc(x)._eval_nseries(x, n, logx)
        if self._trigfunc(0) != 0:
            baseseries -= 1
        baseseries = baseseries.replace(Pow, lambda t, n: t**n/n, simultaneous=False)
        if self._trigfunc(0) != 0:
            baseseries += EulerGamma + log(x)
        return baseseries.subs(x, self.args[0])._eval_nseries(x, n, logx)


class Si(TrigonometricIntegral):
    r"""
    Sine integral.

    Explanation
    ===========

    This function is defined by

    .. math:: \operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.

    It is an entire function.

    Examples
    ========

    >>> from sympy import Si
    >>> from sympy.abc import z

    The sine integral is an antiderivative of $sin(z)/z$:

    >>> Si(z).diff(z)
    sin(z)/z

    It is unbranched:

    >>> from sympy import exp_polar, I, pi
    >>> Si(z*exp_polar(2*I*pi))
    Si(z)

    Sine integral behaves much like ordinary sine under multiplication by ``I``:

    >>> Si(I*z)
    I*Shi(z)
    >>> Si(-z)
    -Si(z)

    It can also be expressed in terms of exponential integrals, but beware
    that the latter is branched:

    >>> from sympy import expint
    >>> Si(z).rewrite(expint)
    -I*(-expint(1, z*exp_polar(-I*pi/2))/2 +
         expint(1, z*exp_polar(I*pi/2))/2) + pi/2

    It can be rewritten in the form of sinc function (by definition):

    >>> from sympy import sinc
    >>> Si(z).rewrite(sinc)
    Integral(sinc(t), (t, 0, z))

    See Also
    ========

    Ci: Cosine integral.
    Shi: Hyperbolic sine integral.
    Chi: Hyperbolic cosine integral.
    Ei: Exponential integral.
    expint: Generalised exponential integral.
    sinc: unnormalized sinc function
    E1: Special case of the generalised exponential integral.
    li: Logarithmic integral.
    Li: Offset logarithmic integral.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral

    """

    _trigfunc = sin
    _atzero = S.Zero

    @classmethod
    def _atinf(cls):
        return pi*S.Half

    @classmethod
    def _atneginf(cls):
        return -pi*S.Half

    @classmethod
    def _minusfactor(cls, z):
        return -Si(z)

    @classmethod
    def _Ifactor(cls, z, sign):
        return I*Shi(z)*sign

    def _eval_rewrite_as_expint(self, z, **kwargs):
        # XXX should we polarify z?
        return pi/2 + (E1(polar_lift(I)*z) - E1(polar_lift(-I)*z))/2/I

    def _eval_rewrite_as_sinc(self, z, **kwargs):
        from sympy.integrals.integrals import Integral
        t = Symbol('t', Dummy=True)
        return Integral(sinc(t), (t, 0, z))

    def _eval_aseries(self, n, args0, x, logx):
        from sympy.series.order import Order
        point = args0[0]

        # Expansion at oo
        if point is S.Infinity:
            z = self.args[0]
            p = [S.NegativeOne**k * factorial(2*k) / z**(2*k)
                    for k in range(0, int((n - 1)/2))] + [Order(1/z**n, x)]
            q = [S.NegativeOne**k * factorial(2*k + 1) / z**(2*k + 1)
                    for k in range(0, int(n/2) - 1)] + [Order(1/z**n, x)]
            return pi/2 - (cos(z)/z)*Add(*p) - (sin(z)/z)*Add(*q)

        # All other points are not handled
        return super(Si, self)._eval_aseries(n, args0, x, logx)

    def _eval_is_zero(self):
        z = self.args[0]
        if z.is_zero:
            return True


class Ci(TrigonometricIntegral):
    r"""
    Cosine integral.

    Explanation
    ===========

    This function is defined for positive $x$ by

    .. math:: \operatorname{Ci}(x) = \gamma + \log{x}
                         + \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t
           = -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t,

    where $\gamma$ is the Euler-Mascheroni constant.

    We have

    .. math:: \operatorname{Ci}(z) =
        -\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right)
               + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}

    which holds for all polar $z$ and thus provides an analytic
    continuation to the Riemann surface of the logarithm.

    The formula also holds as stated
    for $z \in \mathbb{C}$ with $\Re(z) > 0$.
    By lifting to the principal branch, we obtain an analytic function on the
    cut complex plane.

    Examples
    ========

    >>> from sympy import Ci
    >>> from sympy.abc import z

    The cosine integral is a primitive of $\cos(z)/z$:

    >>> Ci(z).diff(z)
    cos(z)/z

    It has a logarithmic branch point at the origin:

    >>> from sympy import exp_polar, I, pi
    >>> Ci(z*exp_polar(2*I*pi))
    Ci(z) + 2*I*pi

    The cosine integral behaves somewhat like ordinary $\cos$ under
    multiplication by $i$:

    >>> from sympy import polar_lift
    >>> Ci(polar_lift(I)*z)
    Chi(z) + I*pi/2
    >>> Ci(polar_lift(-1)*z)
    Ci(z) + I*pi

    It can also be expressed in terms of exponential integrals:

    >>> from sympy import expint
    >>> Ci(z).rewrite(expint)
    -expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2

    See Also
    ========

    Si: Sine integral.
    Shi: Hyperbolic sine integral.
    Chi: Hyperbolic cosine integral.
    Ei: Exponential integral.
    expint: Generalised exponential integral.
    E1: Special case of the generalised exponential integral.
    li: Logarithmic integral.
    Li: Offset logarithmic integral.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral

    """

    _trigfunc = cos
    _atzero = S.ComplexInfinity

    @classmethod
    def _atinf(cls):
        return S.Zero

    @classmethod
    def _atneginf(cls):
        return I*pi

    @classmethod
    def _minusfactor(cls, z):
        return Ci(z) + I*pi

    @classmethod
    def _Ifactor(cls, z, sign):
        return Chi(z) + I*pi/2*sign

    def _eval_rewrite_as_expint(self, z, **kwargs):
        return -(E1(polar_lift(I)*z) + E1(polar_lift(-I)*z))/2

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
        arg0 = arg.subs(x, 0)

        if arg0 is S.NaN:
            arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
        if arg0.is_zero:
            return S.EulerGamma
        elif arg0.is_finite:
            return self.func(arg0)
        else:
            return self

    def _eval_aseries(self, n, args0, x, logx):
        from sympy.series.order import Order
        point = args0[0]

        # Expansion at oo
        if point is S.Infinity:
            z = self.args[0]
            p = [S.NegativeOne**k * factorial(2*k) / z**(2*k)
                    for k in range(0, int((n - 1)/2))] + [Order(1/z**n, x)]
            q = [S.NegativeOne**k * factorial(2*k + 1) / z**(2*k + 1)
                    for k in range(0, int(n/2) - 1)] + [Order(1/z**n, x)]
            return (sin(z)/z)*Add(*p) - (cos(z)/z)*Add(*q)

        # All other points are not handled
        return super(Ci, self)._eval_aseries(n, args0, x, logx)


class Shi(TrigonometricIntegral):
    r"""
    Sinh integral.

    Explanation
    ===========

    This function is defined by

    .. math:: \operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.

    It is an entire function.

    Examples
    ========

    >>> from sympy import Shi
    >>> from sympy.abc import z

    The Sinh integral is a primitive of $\sinh(z)/z$:

    >>> Shi(z).diff(z)
    sinh(z)/z

    It is unbranched:

    >>> from sympy import exp_polar, I, pi
    >>> Shi(z*exp_polar(2*I*pi))
    Shi(z)

    The $\sinh$ integral behaves much like ordinary $\sinh$ under
    multiplication by $i$:

    >>> Shi(I*z)
    I*Si(z)
    >>> Shi(-z)
    -Shi(z)

    It can also be expressed in terms of exponential integrals, but beware
    that the latter is branched:

    >>> from sympy import expint
    >>> Shi(z).rewrite(expint)
    expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2

    See Also
    ========

    Si: Sine integral.
    Ci: Cosine integral.
    Chi: Hyperbolic cosine integral.
    Ei: Exponential integral.
    expint: Generalised exponential integral.
    E1: Special case of the generalised exponential integral.
    li: Logarithmic integral.
    Li: Offset logarithmic integral.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral

    """

    _trigfunc = sinh
    _atzero = S.Zero

    @classmethod
    def _atinf(cls):
        return S.Infinity

    @classmethod
    def _atneginf(cls):
        return S.NegativeInfinity

    @classmethod
    def _minusfactor(cls, z):
        return -Shi(z)

    @classmethod
    def _Ifactor(cls, z, sign):
        return I*Si(z)*sign

    def _eval_rewrite_as_expint(self, z, **kwargs):
        # XXX should we polarify z?
        return (E1(z) - E1(exp_polar(I*pi)*z))/2 - I*pi/2

    def _eval_is_zero(self):
        z = self.args[0]
        if z.is_zero:
            return True

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        arg = self.args[0].as_leading_term(x)
        arg0 = arg.subs(x, 0)

        if arg0 is S.NaN:
            arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
        if arg0.is_zero:
            return arg
        elif not arg0.is_infinite:
            return self.func(arg0)
        elif arg0.is_infinite:
            return -pi*S.ImaginaryUnit/2
        else:
            return self


class Chi(TrigonometricIntegral):
    r"""
    Cosh integral.

    Explanation
    ===========

    This function is defined for positive $x$ by

    .. math:: \operatorname{Chi}(x) = \gamma + \log{x}
                         + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t,

    where $\gamma$ is the Euler-Mascheroni constant.

    We have

    .. math:: \operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right)
                         - i\frac{\pi}{2},

    which holds for all polar $z$ and thus provides an analytic
    continuation to the Riemann surface of the logarithm.
    By lifting to the principal branch we obtain an analytic function on the
    cut complex plane.

    Examples
    ========

    >>> from sympy import Chi
    >>> from sympy.abc import z

    The $\cosh$ integral is a primitive of $\cosh(z)/z$:

    >>> Chi(z).diff(z)
    cosh(z)/z

    It has a logarithmic branch point at the origin:

    >>> from sympy import exp_polar, I, pi
    >>> Chi(z*exp_polar(2*I*pi))
    Chi(z) + 2*I*pi

    The $\cosh$ integral behaves somewhat like ordinary $\cosh$ under
    multiplication by $i$:

    >>> from sympy import polar_lift
    >>> Chi(polar_lift(I)*z)
    Ci(z) + I*pi/2
    >>> Chi(polar_lift(-1)*z)
    Chi(z) + I*pi

    It can also be expressed in terms of exponential integrals:

    >>> from sympy import expint
    >>> Chi(z).rewrite(expint)
    -expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2

    See Also
    ========

    Si: Sine integral.
    Ci: Cosine integral.
    Shi: Hyperbolic sine integral.
    Ei: Exponential integral.
    expint: Generalised exponential integral.
    E1: Special case of the generalised exponential integral.
    li: Logarithmic integral.
    Li: Offset logarithmic integral.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral

    """

    _trigfunc = cosh
    _atzero = S.ComplexInfinity

    @classmethod
    def _atinf(cls):
        return S.Infinity

    @classmethod
    def _atneginf(cls):
        return S.Infinity

    @classmethod
    def _minusfactor(cls, z):
        return Chi(z) + I*pi

    @classmethod
    def _Ifactor(cls, z, sign):
        return Ci(z) + I*pi/2*sign

    def _eval_rewrite_as_expint(self, z, **kwargs):
        return -I*pi/2 - (E1(z) + E1(exp_polar(I*pi)*z))/2

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
        arg0 = arg.subs(x, 0)

        if arg0 is S.NaN:
            arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
        if arg0.is_zero:
            return S.EulerGamma
        elif arg0.is_finite:
            return self.func(arg0)
        else:
            return self


###############################################################################
#################### FRESNEL INTEGRALS ########################################
###############################################################################

class FresnelIntegral(Function):
    """ Base class for the Fresnel integrals."""

    unbranched = True

    @classmethod
    def eval(cls, z):
        # Values at positive infinities signs
        # if any were extracted automatically
        if z is S.Infinity:
            return S.Half

        # Value at zero
        if z.is_zero:
            return S.Zero

        # Try to pull out factors of -1 and I
        prefact = S.One
        newarg = z
        changed = False

        nz = newarg.extract_multiplicatively(-1)
        if nz is not None:
            prefact = -prefact
            newarg = nz
            changed = True

        nz = newarg.extract_multiplicatively(I)
        if nz is not None:
            prefact = cls._sign*I*prefact
            newarg = nz
            changed = True

        if changed:
            return prefact*cls(newarg)

    def fdiff(self, argindex=1):
        if argindex == 1:
            return self._trigfunc(S.Half*pi*self.args[0]**2)
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_is_extended_real(self):
        return self.args[0].is_extended_real

    _eval_is_finite = _eval_is_extended_real

    def _eval_is_zero(self):
        return self.args[0].is_zero

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    as_real_imag = real_to_real_as_real_imag


class fresnels(FresnelIntegral):
    r"""
    Fresnel integral S.

    Explanation
    ===========

    This function is defined by

    .. math:: \operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t.

    It is an entire function.

    Examples
    ========

    >>> from sympy import I, oo, fresnels
    >>> from sympy.abc import z

    Several special values are known:

    >>> fresnels(0)
    0
    >>> fresnels(oo)
    1/2
    >>> fresnels(-oo)
    -1/2
    >>> fresnels(I*oo)
    -I/2
    >>> fresnels(-I*oo)
    I/2

    In general one can pull out factors of -1 and $i$ from the argument:

    >>> fresnels(-z)
    -fresnels(z)
    >>> fresnels(I*z)
    -I*fresnels(z)

    The Fresnel S integral obeys the mirror symmetry
    $\overline{S(z)} = S(\bar{z})$:

    >>> from sympy import conjugate
    >>> conjugate(fresnels(z))
    fresnels(conjugate(z))

    Differentiation with respect to $z$ is supported:

    >>> from sympy import diff
    >>> diff(fresnels(z), z)
    sin(pi*z**2/2)

    Defining the Fresnel functions via an integral:

    >>> from sympy import integrate, pi, sin, expand_func
    >>> integrate(sin(pi*z**2/2), z)
    3*fresnels(z)*gamma(3/4)/(4*gamma(7/4))
    >>> expand_func(integrate(sin(pi*z**2/2), z))
    fresnels(z)

    We can numerically evaluate the Fresnel integral to arbitrary precision
    on the whole complex plane:

    >>> fresnels(2).evalf(30)
    0.343415678363698242195300815958

    >>> fresnels(-2*I).evalf(30)
    0.343415678363698242195300815958*I

    See Also
    ========

    fresnelc: Fresnel cosine integral.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Fresnel_integral
    .. [2] http://dlmf.nist.gov/7
    .. [3] http://mathworld.wolfram.com/FresnelIntegrals.html
    .. [4] http://functions.wolfram.com/GammaBetaErf/FresnelS
    .. [5] The converging factors for the fresnel integrals
            by John W. Wrench Jr. and Vicki Alley

    """
    _trigfunc = sin
    _sign = -S.One

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0:
            return S.Zero
        else:
            x = sympify(x)
            if len(previous_terms) > 1:
                p = previous_terms[-1]
                return (-pi**2*x**4*(4*n - 1)/(8*n*(2*n + 1)*(4*n + 3))) * p
            else:
                return x**3 * (-x**4)**n * (S(2)**(-2*n - 1)*pi**(2*n + 1)) / ((4*n + 3)*factorial(2*n + 1))

    def _eval_rewrite_as_erf(self, z, **kwargs):
        return (S.One + I)/4 * (erf((S.One + I)/2*sqrt(pi)*z) - I*erf((S.One - I)/2*sqrt(pi)*z))

    def _eval_rewrite_as_hyper(self, z, **kwargs):
        return pi*z**3/6 * hyper([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)], -pi**2*z**4/16)

    def _eval_rewrite_as_meijerg(self, z, **kwargs):
        return (pi*z**Rational(9, 4) / (sqrt(2)*(z**2)**Rational(3, 4)*(-z)**Rational(3, 4))
                * meijerg([], [1], [Rational(3, 4)], [Rational(1, 4), 0], -pi**2*z**4/16))

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy.series.order import Order
        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
        arg0 = arg.subs(x, 0)

        if arg0 is S.ComplexInfinity:
            arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
        if arg0.is_zero:
            return pi*arg**3/6
        elif arg0 in [S.Infinity, S.NegativeInfinity]:
            s = 1 if arg0 is S.Infinity else -1
            return s*S.Half + Order(x, x)
        else:
            return self.func(arg0)

    def _eval_aseries(self, n, args0, x, logx):
        from sympy.series.order import Order
        point = args0[0]

        # Expansion at oo and -oo
        if point in [S.Infinity, -S.Infinity]:
            z = self.args[0]

            # expansion of S(x) = S1(x*sqrt(pi/2)), see reference[5] page 1-8
            # as only real infinities are dealt with, sin and cos are O(1)
            p = [S.NegativeOne**k * factorial(4*k + 1) /
                 (2**(2*k + 2) * z**(4*k + 3) * 2**(2*k)*factorial(2*k))
                 for k in range(0, n) if 4*k + 3 < n]
            q = [1/(2*z)] + [S.NegativeOne**k * factorial(4*k - 1) /
                 (2**(2*k + 1) * z**(4*k + 1) * 2**(2*k - 1)*factorial(2*k - 1))
                 for k in range(1, n) if 4*k + 1 < n]

            p = [-sqrt(2/pi)*t for t in p]
            q = [-sqrt(2/pi)*t for t in q]
            s = 1 if point is S.Infinity else -1
            # The expansion at oo is 1/2 + some odd powers of z
            # To get the expansion at -oo, replace z by -z and flip the sign
            # The result -1/2 + the same odd powers of z as before.
            return s*S.Half + (sin(z**2)*Add(*p) + cos(z**2)*Add(*q)
                ).subs(x, sqrt(2/pi)*x) + Order(1/z**n, x)

        # All other points are not handled
        return super()._eval_aseries(n, args0, x, logx)


class fresnelc(FresnelIntegral):
    r"""
    Fresnel integral C.

    Explanation
    ===========

    This function is defined by

    .. math:: \operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t.

    It is an entire function.

    Examples
    ========

    >>> from sympy import I, oo, fresnelc
    >>> from sympy.abc import z

    Several special values are known:

    >>> fresnelc(0)
    0
    >>> fresnelc(oo)
    1/2
    >>> fresnelc(-oo)
    -1/2
    >>> fresnelc(I*oo)
    I/2
    >>> fresnelc(-I*oo)
    -I/2

    In general one can pull out factors of -1 and $i$ from the argument:

    >>> fresnelc(-z)
    -fresnelc(z)
    >>> fresnelc(I*z)
    I*fresnelc(z)

    The Fresnel C integral obeys the mirror symmetry
    $\overline{C(z)} = C(\bar{z})$:

    >>> from sympy import conjugate
    >>> conjugate(fresnelc(z))
    fresnelc(conjugate(z))

    Differentiation with respect to $z$ is supported:

    >>> from sympy import diff
    >>> diff(fresnelc(z), z)
    cos(pi*z**2/2)

    Defining the Fresnel functions via an integral:

    >>> from sympy import integrate, pi, cos, expand_func
    >>> integrate(cos(pi*z**2/2), z)
    fresnelc(z)*gamma(1/4)/(4*gamma(5/4))
    >>> expand_func(integrate(cos(pi*z**2/2), z))
    fresnelc(z)

    We can numerically evaluate the Fresnel integral to arbitrary precision
    on the whole complex plane:

    >>> fresnelc(2).evalf(30)
    0.488253406075340754500223503357

    >>> fresnelc(-2*I).evalf(30)
    -0.488253406075340754500223503357*I

    See Also
    ========

    fresnels: Fresnel sine integral.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Fresnel_integral
    .. [2] http://dlmf.nist.gov/7
    .. [3] http://mathworld.wolfram.com/FresnelIntegrals.html
    .. [4] http://functions.wolfram.com/GammaBetaErf/FresnelC
    .. [5] The converging factors for the fresnel integrals
            by John W. Wrench Jr. and Vicki Alley

    """
    _trigfunc = cos
    _sign = S.One

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0:
            return S.Zero
        else:
            x = sympify(x)
            if len(previous_terms) > 1:
                p = previous_terms[-1]
                return (-pi**2*x**4*(4*n - 3)/(8*n*(2*n - 1)*(4*n + 1))) * p
            else:
                return x * (-x**4)**n * (S(2)**(-2*n)*pi**(2*n)) / ((4*n + 1)*factorial(2*n))

    def _eval_rewrite_as_erf(self, z, **kwargs):
        return (S.One - I)/4 * (erf((S.One + I)/2*sqrt(pi)*z) + I*erf((S.One - I)/2*sqrt(pi)*z))

    def _eval_rewrite_as_hyper(self, z, **kwargs):
        return z * hyper([Rational(1, 4)], [S.Half, Rational(5, 4)], -pi**2*z**4/16)

    def _eval_rewrite_as_meijerg(self, z, **kwargs):
        return (pi*z**Rational(3, 4) / (sqrt(2)*root(z**2, 4)*root(-z, 4))
                * meijerg([], [1], [Rational(1, 4)], [Rational(3, 4), 0], -pi**2*z**4/16))

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy.series.order import Order
        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
        arg0 = arg.subs(x, 0)

        if arg0 is S.ComplexInfinity:
            arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
        if arg0.is_zero:
            return arg
        elif arg0 in [S.Infinity, S.NegativeInfinity]:
            s = 1 if arg0 is S.Infinity else -1
            return s*S.Half + Order(x, x)
        else:
            return self.func(arg0)

    def _eval_aseries(self, n, args0, x, logx):
        from sympy.series.order import Order
        point = args0[0]

        # Expansion at oo
        if point in [S.Infinity, -S.Infinity]:
            z = self.args[0]

            # expansion of C(x) = C1(x*sqrt(pi/2)), see reference[5] page 1-8
            # as only real infinities are dealt with, sin and cos are O(1)
            p = [S.NegativeOne**k * factorial(4*k + 1) /
                 (2**(2*k + 2) * z**(4*k + 3) * 2**(2*k)*factorial(2*k))
                 for k in range(0, n) if 4*k + 3 < n]
            q = [1/(2*z)] + [S.NegativeOne**k * factorial(4*k - 1) /
                 (2**(2*k + 1) * z**(4*k + 1) * 2**(2*k - 1)*factorial(2*k - 1))
                 for k in range(1, n) if 4*k + 1 < n]

            p = [-sqrt(2/pi)*t for t in p]
            q = [ sqrt(2/pi)*t for t in q]
            s = 1 if point is S.Infinity else -1
            # The expansion at oo is 1/2 + some odd powers of z
            # To get the expansion at -oo, replace z by -z and flip the sign
            # The result -1/2 + the same odd powers of z as before.
            return s*S.Half + (cos(z**2)*Add(*p) + sin(z**2)*Add(*q)
                ).subs(x, sqrt(2/pi)*x) + Order(1/z**n, x)

        # All other points are not handled
        return super()._eval_aseries(n, args0, x, logx)


###############################################################################
#################### HELPER FUNCTIONS #########################################
###############################################################################


class _erfs(Function):
    """
    Helper function to make the $\\mathrm{erf}(z)$ function
    tractable for the Gruntz algorithm.

    """
    @classmethod
    def eval(cls, arg):
        if arg.is_zero:
            return S.One

    def _eval_aseries(self, n, args0, x, logx):
        from sympy.series.order import Order
        point = args0[0]

        # Expansion at oo
        if point is S.Infinity:
            z = self.args[0]
            l = [ 1/sqrt(S.Pi) * factorial(2*k)*(-S(
                4))**(-k)/factorial(k) * (1/z)**(2*k + 1) for k in range(0, n) ]
            o = Order(1/z**(2*n + 1), x)
            # It is very inefficient to first add the order and then do the nseries
            return (Add(*l))._eval_nseries(x, n, logx) + o

        # Expansion at I*oo
        t = point.extract_multiplicatively(S.ImaginaryUnit)
        if t is S.Infinity:
            z = self.args[0]
            # TODO: is the series really correct?
            l = [ 1/sqrt(S.Pi) * factorial(2*k)*(-S(
                4))**(-k)/factorial(k) * (1/z)**(2*k + 1) for k in range(0, n) ]
            o = Order(1/z**(2*n + 1), x)
            # It is very inefficient to first add the order and then do the nseries
            return (Add(*l))._eval_nseries(x, n, logx) + o

        # All other points are not handled
        return super()._eval_aseries(n, args0, x, logx)

    def fdiff(self, argindex=1):
        if argindex == 1:
            z = self.args[0]
            return -2/sqrt(S.Pi) + 2*z*_erfs(z)
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_rewrite_as_intractable(self, z, **kwargs):
        return (S.One - erf(z))*exp(z**2)


class _eis(Function):
    """
    Helper function to make the $\\mathrm{Ei}(z)$ and $\\mathrm{li}(z)$
    functions tractable for the Gruntz algorithm.

    """


    def _eval_aseries(self, n, args0, x, logx):
        from sympy.series.order import Order
        if args0[0] != S.Infinity:
            return super(_erfs, self)._eval_aseries(n, args0, x, logx)

        z = self.args[0]
        l = [ factorial(k) * (1/z)**(k + 1) for k in range(0, n) ]
        o = Order(1/z**(n + 1), x)
        # It is very inefficient to first add the order and then do the nseries
        return (Add(*l))._eval_nseries(x, n, logx) + o


    def fdiff(self, argindex=1):
        if argindex == 1:
            z = self.args[0]
            return S.One / z - _eis(z)
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_rewrite_as_intractable(self, z, **kwargs):
        return exp(-z)*Ei(z)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        x0 = self.args[0].limit(x, 0)
        if x0.is_zero:
            f = self._eval_rewrite_as_intractable(*self.args)
            return f._eval_as_leading_term(x, logx=logx, cdir=cdir)
        return super()._eval_as_leading_term(x, logx=logx, cdir=cdir)

    def _eval_nseries(self, x, n, logx, cdir=0):
        x0 = self.args[0].limit(x, 0)
        if x0.is_zero:
            f = self._eval_rewrite_as_intractable(*self.args)
            return f._eval_nseries(x, n, logx)
        return super()._eval_nseries(x, n, logx)
