gdZddlmZmZddlmZddlmZmZGddeZ Gdde Z Gd d e Z Gd d e Z Gd de Z dS)z- This module contains the Mathieu functions. )FunctionArgumentIndexError)sqrt)sincosceZdZdZdZdZdS) MathieuBasezj Abstract base class for Mathieu functions. This class is meant to reduce code duplication. Tc|j\}}}||||SN)argsfunc conjugate)selfaqzs e/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/functions/special/mathieu_functions.py_eval_conjugatezMathieuBase._eval_conjugates=)1ayy q{{}}EEEN)__name__ __module__ __qualname____doc__ unbranchedrrrr r s9JFFFFFrr c0eZdZdZddZedZdS)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/ cf|dkr|j\}}}t|||St||N)r mathieusprimerrargindexrrrs rfdiffzmathieus.fdiffF: q==iGAq! Aq)) )$T844 4rc|jr&|jrtt||zS|r||||  SdSr ) is_Numberis_zerorrcould_extract_minus_signclsrrrs revalz mathieus.evalMs_ ; "19 "tAwwqy>> ! % % ' ' "C1qbMM> ! " "rNrrrrrr% classmethodr-rrrrrN++Z5555""["""rrc0eZdZdZddZedZdS)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/ rcf|dkr|j\}}}t|||St||r )r mathieucprimerr#s rr%zmathieuc.fdiffr&rc|jr&|jrtt||zS|r|||| SdSr )r(r)rrr*r+s rr-z mathieuc.evals] ; "19 "tAwwqy>> ! % % ' ' !3q!aR==  ! !rNr.r/rrrr3r3VN++Z5555!![!!!rr3c0eZdZdZddZedZdS)r"a" 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/ rc|dkr8|j\}}}d|ztd|zz|z t|||zSt||Nr!)r rrrr#s rr%zmathieusprime.fdiffV q==iGAq!aCAaCL1$hq!Q&7&77 7$T844 4rc|jr6|jr/t|tt||zzS|r|||| SdSr )r(r)rrr*r+s rr-zmathieusprime.evalsh ; *19 *773tAwwqy>>) ) % % ' ' !3q!aR==  ! !rNr.r/rrrr"r"r7rr"c0eZdZdZddZedZdS)r5a! 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/ rc|dkr8|j\}}}d|ztd|zz|z t|||zSt||r:)r rr3rr#s rr%zmathieucprime.fdiffr<rc|jr7|jr0t| tt||zzS|r||||  SdSr )r(r)rrr*r+s rr-zmathieucprime.evalsl ; +19 +GG8CQ NN* * % % ' ' "C1qbMM> ! 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