U L?h@sdZddlmZmZddlmZddlmZmZGdddeZ Gddde Z Gd d d e Z Gd d d e Z Gd dde Z dS)z- This module contains the Mathieu functions. )FunctionArgumentIndexError)sqrt)sincosc@seZdZdZdZddZdS) MathieuBasezj Abstract base class for Mathieu functions. This class is meant to reduce code duplication. TcCs&|j\}}}||||SN)argsfunc conjugate)selfaqzr[/var/www/html/venv/lib/python3.8/site-packages/sympy/functions/special/mathieu_functions.py_eval_conjugates zMathieuBase._eval_conjugateN)__name__ __module__ __qualname____doc__Z unbranchedrrrrrr src@s&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/ cCs.|dkr |j\}}}t|||St||dSN)r mathieusprimerr Zargindexr rrrrrfdiffFs  zmathieus.fdiffcCs8|jr|jrtt||S|r4||||  SdSr) is_Numberis_zerorrcould_extract_minus_signclsr rrrrrevalMs z mathieus.evalN)rrrrrr classmethodr#rrrrrs- rc@s&eZdZdZdddZeddZdS) 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/ rcCs.|dkr |j\}}}t|||St||dSr)r mathieucprimerrrrrrs  zmathieuc.fdiffcCs6|jr|jrtt||S|r2|||| SdSr)rrrrr r!rrrr#s z mathieuc.evalN)rr$rrrrr&Vs- r&c@s&eZdZdZdddZeddZdS) 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/ rcCsF|dkr8|j\}}}d|td||t|||St||dSNr)r rrrrrrrrs $zmathieusprime.fdiffcCs>|jr$|jr$t|tt||S|r:|||| SdSr)rrrrr r!rrrr#s zmathieusprime.evalN)rr$rrrrrs- rc@s&eZdZdZdddZeddZdS) 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/ rcCsF|dkr8|j\}}}d|td||t|||St||dSr()r rr&rrrrrrs $zmathieucprime.fdiffcCsB|jr&|jr&t| tt||S|r>||||  SdSr)rrrrr r!rrrr#s zmathieucprime.evalN)rr$rrrrr's- r'N)rZsympy.core.functionrrZ(sympy.functions.elementary.miscellaneousrZ(sympy.functions.elementary.trigonometricrrrrr&rr'rrrrs >>>