# Source code for sympy.functions.special.error_functions

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

from sympy.core import Add, S, C, sympify, cacheit, pi, I
from sympy.core.function import Function, ArgumentIndexError
from sympy.functions.elementary.miscellaneous import sqrt, root
from sympy.functions.elementary.complexes import polar_lift
from sympy.functions.special.hyper import hyper, meijerg

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

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

[docs]class erf(Function):
"""
The Gauss error function.

This function is defined as:

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

Or, in ASCII::

x
/
|
|     2
|   -t
2* |  e    dt
|
/
0
-------------
____
\/ pi

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

References
==========

.. [1] http://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
"""

nargs = 1
unbranched = True

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

@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 S.Zero:
return S.Zero

t = arg.extract_multiplicatively(S.ImaginaryUnit)
if t == S.Infinity or t == S.NegativeInfinity:
return arg

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 = C.floor((n - 1)/S(2))
if len(previous_terms) > 2:
return -previous_terms[-2] * x**2 * (n-2)/(n*k)
else:
return 2*(-1)**k * x**n/(n*C.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_real

def _eval_rewrite_as_uppergamma(self, z):
return sqrt(z**2)/z*(S.One - C.uppergamma(S.Half, z**2)/sqrt(S.Pi))

def _eval_rewrite_as_tractable(self, z):
return S.One - _erfs(z)*C.exp(-z**2)

if x in arg.free_symbols and C.Order(1, x).contains(arg):
return 2*x/sqrt(pi)
else:
return self.func(arg)

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

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

For the 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 :math:\gamma is the Euler-Mascheroni constant.

If :math: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 :math:\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 :math:x > 0 and :math:\operatorname{Ei}(x) as defined above.

Note that we carefully avoided defining :math:\operatorname{Ei}(x) for
negative real x. This is because above integral formula does not hold for
any polar lift of such :math:x, indeed all branches of
:math:\operatorname{Ei}(x) above the negative reals are imaginary.

However, the following statement holds for all :math:x \in \mathbb{R}^*:

.. math:: \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t =
\frac{\operatorname{Ei}\left(|x|e^{i \arg(x)}\right) +
\operatorname{Ei}\left(|x|e^{- i \arg(x)}\right)}{2},

where the integral is again understood to be a principal value if
:math:x > 0, and :math:|x|e^{i \arg(x)},
:math:|x|e^{- i \arg(x)} denote two conjugate polar lifts of :math:x.

========

expint, sympy.functions.special.gamma_functions.uppergamma

References
==========

- Abramowitz & Stegun, section 5: http://www.math.sfu.ca/~cbm/aands/page_228.htm
- http://en.wikipedia.org/wiki/Exponential_integral

Examples
========

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

The exponential integral in SymPy is strictly undefined for negative values
of the argument. For convenience, exponential integrals with negative
arguments are immediately converted into an expression that agrees with
the classical integral definition:

>>> Ei(-1)
-I*pi + Ei(exp_polar(I*pi))

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

"""

nargs = 1

@classmethod
def eval(cls, z):
if not z.is_polar and z.is_negative:
# Note: is this a good idea?
return Ei(polar_lift(z)) - pi*I
nz, n = z.extract_branch_factor()
if n:
return Ei(nz) + 2*I*pi*n

def fdiff(self, argindex=1):
from sympy import unpolarify
arg = unpolarify(self.args[0])
if argindex == 1:
return C.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):
from sympy 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):
return -expint(1, polar_lift(-1)*z) - I*pi

def _eval_rewrite_as_Si(self, z):
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

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

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 :math:z with positive real part we have

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

which explains the name.

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

========

E1: The classical case, returns expint(1, z).
Ei: Another related function called exponential integral.
sympy.functions.special.gamma_functions.uppergamma

References
==========

- http://dlmf.nist.gov/8.19
- http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/
- http://en.wikipedia.org/wiki/Exponential_integral

Examples
========

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

Differentiation is supported. Differentiation with respect to z explains
further 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, -nu + 1), ()), 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)*erf(sqrt(z))/sqrt(z) + sqrt(pi)/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(-nu + 1, 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(-nu + 1) + expint(nu, z)

"""

nargs = 2

@classmethod
def eval(cls, nu, z):
from sympy import (unpolarify, expand_mul, uppergamma, exp, gamma,
factorial)
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 == 0:
return
if nu.is_integer:
if (nu > 0) is not True:
return
return expint(nu, z) \
- 2*pi*I*n*(-1)**(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):
from sympy import meijerg
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):
from sympy import uppergamma
return z**(nu - 1)*uppergamma(1 - nu, z)

def _eval_rewrite_as_Ei(self, nu, z):
from sympy import exp_polar, unpolarify, exp, factorial
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):
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

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

This is equivalent to expint(1, z).

"""
return expint(1, z)

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

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

nargs = 1

@classmethod
def eval(cls, z):
if z == 0:
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):
from sympy import unpolarify
arg = unpolarify(self.args[0])
if argindex == 1:
return self._trigfunc(arg)/arg

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

def _eval_rewrite_as_uppergamma(self, z):
from sympy import uppergamma
return self._eval_rewrite_as_expint(z).rewrite(uppergamma)

def _eval_nseries(self, x, n, logx):
# NOTE this is fairly inefficient
from sympy import log, EulerGamma, Pow
n += 1
if self.args[0].subs(x, 0) != 0:
return super(TrigonometricIntegral, self)._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)
if self._trigfunc(0) != 0:
baseseries += EulerGamma + log(x)
return baseseries.subs(x, self.args[0])._eval_nseries(x, n, logx)

[docs]class Si(TrigonometricIntegral):
r"""
Sine integral.

This function is defined by

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

It is an entire function.

========

Ci: Cosine integral.
Shi: Sinh integral.
Chi: Cosh integral.
expint: The generalised exponential integral.

References
==========

- http://en.wikipedia.org/wiki/Trigonometric_integral

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

"""

_trigfunc = C.sin
_atzero = S(0)

@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):
# XXX should we polarify z?
return pi/2 + (E1(polar_lift(I)*z) - E1(polar_lift(-I)*z))/2/I

[docs]class Ci(TrigonometricIntegral):
r"""
Cosine integral.

This function is defined for positive :math: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 :math:\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 :math:z and thus provides an analytic
continuation to the Riemann surface of the logarithm.

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

========

Si: Sine integral.
Shi: Sinh integral.
Chi: Cosh integral.
expint: The generalised exponential integral.

References
==========

- http://en.wikipedia.org/wiki/Trigonometric_integral

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

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

"""

_trigfunc = C.cos
_atzero = S.ComplexInfinity

@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):
return -(E1(polar_lift(I)*z) + E1(polar_lift(-I)*z))/2

[docs]class Shi(TrigonometricIntegral):
r"""
Sinh integral.

This function is defined by

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

It is an entire function.

========

Si: Sine integral.
Ci: Cosine integral.
Chi: Cosh integral.
expint: The generalised exponential integral.

References
==========

- http://en.wikipedia.org/wiki/Trigonometric_integral

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)

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

"""

_trigfunc = C.sinh
_atzero = S(0)

@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):
from sympy import exp_polar
# XXX should we polarify z?
return (E1(z)-E1(exp_polar(I*pi)*z))/2 - I*pi/2

[docs]class Chi(TrigonometricIntegral):
r"""
Cosh integral.

This function is defined for positive :math:x by

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

where :math:\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 :math: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.

========

Si: Sine integral.
Ci: Cosine integral.
Shi: Sinh integral.
expint: The generalised exponential integral.

References
==========

- http://en.wikipedia.org/wiki/Trigonometric_integral

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

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

"""

_trigfunc = C.cosh
_atzero = S.ComplexInfinity

@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):
from sympy import exp_polar
return -I*pi/2 - (E1(z) + E1(exp_polar(I*pi)*z))/2

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

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

nargs = 1
unbranched = True

@classmethod
def eval(cls, z):
# Value at zero
if z is S.Zero:
return S(0)

# 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)

# Values at positive infinities signs
# if any were extracted automatically
if z is S.Infinity:
return S.Half
elif z is I*S.Infinity:
return cls._sign*I*S.Half

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_real(self):
return self.args[0].is_real

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

def _as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (re, im)

def as_real_imag(self, deep=True, **hints):
# Fresnel S
# http://functions.wolfram.com/06.32.19.0003.01
# http://functions.wolfram.com/06.32.19.0006.01
# Fresnel C
# http://functions.wolfram.com/06.33.19.0003.01
# http://functions.wolfram.com/06.33.19.0006.01
x, y = self._as_real_imag(deep=deep, **hints)
sq = -y**2/x**2
re = S.Half*(self.func(x+x*sqrt(sq))+self.func(x-x*sqrt(sq)))
im = x/(2*y) * sqrt(sq) * (self.func(x-x*sqrt(sq)) - self.func(x+x*sqrt(sq)))
return (re, im)

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

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:

>>> 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, gamma, 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

========

fresnelc

References
==========

.. [1] http://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
"""

_trigfunc = C.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)*C.factorial(2*n + 1))

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

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

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

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:

>>> 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, gamma, 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

========

fresnels

References
==========

.. [1] http://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
"""

_trigfunc = C.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)*C.factorial(2*n))

def _eval_rewrite_as_erf(self, z):
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):
return z * hyper([S.One/4], [S.One/2, S(5)/4], -pi**2*z**4/16)

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

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

class _erfs(Function):
"""
Helper function to make the :math:erf(z) function
tractable for the Gruntz algorithm.
"""

nargs = 1

def _eval_aseries(self, n, args0, x, logx):
if args0[0] != S.Infinity:
return super(_erfs, self)._eval_aseries(n, args0, x, logx)

z = self.args[0]
l = [ 1/sqrt(S.Pi) * C.factorial(2*k)*(-S(4))**(-k)/C.factorial(k) * (1/z)**(2*k+1) for k in xrange(0,n) ]
o = C.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

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):
return (S.One-erf(z))*C.exp(z**2)