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