o i@sdZddlmZmZddlmZddlmZmZGdddeZ Gddde Z Gd d d e Z Gd d d e Z Gd dde Z dS)z- This module contains the Mathieu functions. )DefinedFunctionArgumentIndexError)sqrt)sincosc@seZdZdZdZddZdS) MathieuBasezj Abstract base class for Mathieu functions. This class is meant to reduce code duplication. TcCs&|j\}}}||||SN)argsfunc conjugate)selfaqzrx/var/www/html/eduruby.in/lip-sync/lip-sync-env/lib/python3.10/site-packages/sympy/functions/special/mathieu_functions.py_eval_conjugates zMathieuBase._eval_conjugateN)__name__ __module__ __qualname____doc__Z unbranchedrrrrrr s rc@&eZdZdZdddZeddZdS) mathieusa 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/ cC*|dkr|j\}}}t|||St||N)r mathieusprimerr Zargindexr rrrrrfdiffF   zmathieus.fdiffcCs8|jr|jrtt||S|r||||  SdSr) is_Numberis_zerorrcould_extract_minus_signclsr rrrrrevalMs z mathieus.evalNrrrrrr classmethodr&rrrrr  -rc@r) mathieuca 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/ rcCrr)r mathieucprimerrrrrrr zmathieuc.fdiffcCs6|jr|jrtt||S|r|||| SdSr)r!r"rrr#r$rrrr&s z mathieuc.evalNr'r(rrrrr+Vr*r+c@r) ra" 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/ rcCB|dkr|j\}}}d|td||t|||St||Nr)r rrrrrrrr $ zmathieusprime.fdiffcCs>|jr|jrt|tt||S|r|||| SdSr)r!r"rrr#r$rrrr&s zmathieusprime.evalNr'r(rrrrrr*rc@r) r,a! 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/ rcCr-r.)r rr+rrrrrrr0zmathieucprime.fdiffcCsB|jr|jrt| tt||S|r||||  SdSr)r!r"rrr#r$rrrr&s zmathieucprime.evalNr'r(rrrrr,r*r,N)rZsympy.core.functionrrZ(sympy.functions.elementary.miscellaneousrZ(sympy.functions.elementary.trigonometricrrrrr+rr,rrrrs >>>