/

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) See Also ======== 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) See Also ======== 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) See Also ======== 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) See Also ======== 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)