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