# Source code for sympy.functions.special.mathieu_functions

""" This module contains the Mathieu functions.
"""

from __future__ import print_function, division

from sympy.core import S
from sympy.core.function import Function, ArgumentIndexError
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import sin, cos

[docs]class MathieuBase(Function):
"""
Abstract base class for Mathieu functions.

This class is meant to reduce code duplication.
"""

unbranched = True

def _eval_conjugate(self):
a, q, z = self.args
return self.func(a.conjugate(), q.conjugate(), z.conjugate())

[docs]class mathieus(MathieuBase):
r"""
The Mathieu Sine function S(a,q,z). This function is one solution
of the Mathieu differential equation:

.. math ::
y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0

The other solution is the Mathieu Cosine function.

Examples
========

>>> from sympy import diff, mathieus
>>> from sympy.abc import a, q, z

>>> mathieus(a, q, z)
mathieus(a, q, z)

>>> mathieus(a, 0, z)
sin(sqrt(a)*z)

>>> diff(mathieus(a, q, z), z)
mathieusprime(a, q, z)

========

mathieuc: Mathieu cosine function.
mathieusprime: Derivative of Mathieu sine function.
mathieucprime: Derivative of Mathieu cosine function.

References
==========

.. [1] http://en.wikipedia.org/wiki/Mathieu_function
.. [2] http://dlmf.nist.gov/28
.. [3] http://mathworld.wolfram.com/MathieuBase.html
.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuS/
"""

def fdiff(self, argindex=1):
if argindex == 3:
a, q, z = self.args
return mathieusprime(a, q, z)
else:
raise ArgumentIndexError(self, argindex)

@classmethod
def eval(cls, a, q, z):
if q.is_Number and q is S.Zero:
return sin(sqrt(a)*z)
# Try to pull out factors of -1
if z.could_extract_minus_sign():
return -cls(a, q, -z)

[docs]class mathieuc(MathieuBase):
r"""
The Mathieu Cosine function C(a,q,z). This function is one solution
of the Mathieu differential equation:

.. math ::
y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0

The other solution is the Mathieu Sine function.

Examples
========

>>> from sympy import diff, mathieuc
>>> from sympy.abc import a, q, z

>>> mathieuc(a, q, z)
mathieuc(a, q, z)

>>> mathieuc(a, 0, z)
cos(sqrt(a)*z)

>>> diff(mathieuc(a, q, z), z)
mathieucprime(a, q, z)

========

mathieus: Mathieu sine function
mathieusprime: Derivative of Mathieu sine function
mathieucprime: Derivative of Mathieu cosine function

References
==========

.. [1] http://en.wikipedia.org/wiki/Mathieu_function
.. [2] http://dlmf.nist.gov/28
.. [3] http://mathworld.wolfram.com/MathieuBase.html
.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuC/
"""

def fdiff(self, argindex=1):
if argindex == 3:
a, q, z = self.args
return mathieucprime(a, q, z)
else:
raise ArgumentIndexError(self, argindex)

@classmethod
def eval(cls, a, q, z):
if q.is_Number and q is S.Zero:
return cos(sqrt(a)*z)
# Try to pull out factors of -1
if z.could_extract_minus_sign():
return cls(a, q, -z)

[docs]class mathieusprime(MathieuBase):
r"""
The derivative S^{\prime}(a,q,z) of the Mathieu Sine function.
This function is one solution of the Mathieu differential equation:

.. math ::
y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0

The other solution is the Mathieu Cosine function.

Examples
========

>>> from sympy import diff, mathieusprime
>>> from sympy.abc import a, q, z

>>> mathieusprime(a, q, z)
mathieusprime(a, q, z)

>>> mathieusprime(a, 0, z)
sqrt(a)*cos(sqrt(a)*z)

>>> diff(mathieusprime(a, q, z), z)
(-a + 2*q*cos(2*z))*mathieus(a, q, z)

========

mathieus: Mathieu sine function
mathieuc: Mathieu cosine function
mathieucprime: Derivative of Mathieu cosine function

References
==========

.. [1] http://en.wikipedia.org/wiki/Mathieu_function
.. [2] http://dlmf.nist.gov/28
.. [3] http://mathworld.wolfram.com/MathieuBase.html
.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuSPrime/
"""

def fdiff(self, argindex=1):
if argindex == 3:
a, q, z = self.args
return (2*q*cos(2*z) - a)*mathieus(a, q, z)
else:
raise ArgumentIndexError(self, argindex)

@classmethod
def eval(cls, a, q, z):
if q.is_Number and q is S.Zero:
return sqrt(a)*cos(sqrt(a)*z)
# Try to pull out factors of -1
if z.could_extract_minus_sign():
return cls(a, q, -z)

[docs]class mathieucprime(MathieuBase):
r"""
The derivative C^{\prime}(a,q,z) of the Mathieu Cosine function.
This function is one solution of the Mathieu differential equation:

.. math ::
y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0

The other solution is the Mathieu Sine function.

Examples
========

>>> from sympy import diff, mathieucprime
>>> from sympy.abc import a, q, z

>>> mathieucprime(a, q, z)
mathieucprime(a, q, z)

>>> mathieucprime(a, 0, z)
-sqrt(a)*sin(sqrt(a)*z)

>>> diff(mathieucprime(a, q, z), z)
(-a + 2*q*cos(2*z))*mathieuc(a, q, z)

========

mathieus: Mathieu sine function
mathieuc: Mathieu cosine function
mathieusprime: Derivative of Mathieu sine function

References
==========

.. [1] http://en.wikipedia.org/wiki/Mathieu_function
.. [2] http://dlmf.nist.gov/28
.. [3] http://mathworld.wolfram.com/MathieuBase.html
.. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuCPrime/
"""

def fdiff(self, argindex=1):
if argindex == 3:
a, q, z = self.args
return (2*q*cos(2*z) - a)*mathieuc(a, q, z)
else:
raise ArgumentIndexError(self, argindex)

@classmethod
def eval(cls, a, q, z):
if q.is_Number and q is S.Zero:
return -sqrt(a)*sin(sqrt(a)*z)
# Try to pull out factors of -1
if z.could_extract_minus_sign():
return -cls(a, q, -z)