7^(hdZddlmZmZddlmZddlmZmZGddeZ Gdde Z Gd d e Z Gd d e Z Gd de Z y)z- This module contains the Mathieu functions. )DefinedFunctionArgumentIndexError)sqrt)sincosceZdZdZdZdZy) MathieuBasezj Abstract base class for Mathieu functions. This class is meant to reduce code duplication. Tc|j\}}}|j|j|j|jSN)argsfunc conjugate)selfaqzs g/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/sympy/functions/special/mathieu_functions.py_eval_conjugatezMathieuBase._eval_conjugates6))1ayy q{{}EEN)__name__ __module__ __qualname____doc__ unbranchedrrrr r sJFrr c(eZdZdZddZedZy)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/ c^|dk(r|j\}}}t|||St||N)r mathieusprimerrargindexrrrs rfdiffzmathieus.fdiffF4 q=iiGAq! Aq) )$T84 4rc|jr#|jrtt||zS|j r ||||  Syr ) is_Numberis_zerorrcould_extract_minus_signclsrrrs revalz mathieus.evalMsD ;;199tAwqy> ! % % '1qbM> ! (rNrrrrr$ classmethodr,rrrrr!+Z5""rrc(eZdZdZddZedZy)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/ c^|dk(r|j\}}}t|||St||r)r mathieucprimerr"s rr$zmathieuc.fdiffr%rc|jr#|jrtt||zS|j r |||| Syr )r'r(rrr)r*s rr,z mathieuc.evalsB ;;199tAwqy> ! % % 'q!aR=  (rNr-r/rrrr3r3V!+Z5!!rr3c(eZdZdZddZedZy)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/ c|dk(r5|j\}}}d|ztd|zz|z t|||zSt||Nr )r rrrr"s rr$zmathieusprime.fdiffN q=iiGAq!aCAaCL1$hq!Q&77 7$T84 4rc|jr/|jr#t|tt||zzS|j r |||| Syr )r'r(rrr)r*s rr,zmathieusprime.evalsK ;;19973tAwqy>) ) % % 'q!aR=  (rNr-r/rrrr!r!r7rr!c(eZdZdZddZedZy)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/ c|dk(r5|j\}}}d|ztd|zz|z t|||zSt||r:)r rr3rr"s rr$zmathieucprime.fdiffr<rc|jr0|jr$t| tt||zzS|j r ||||  Syr )r'r(rrr)r*s rr,zmathieucprime.evalsO ;;199G8CQ N* * % % '1qbM> ! (rNr-r/rrrr5r5r1rr5N)rsympy.core.functionrr(sympy.functions.elementary.miscellaneousr(sympy.functions.elementary.trigonometricrrr rr3r!r5rrrrDsWD9= F/ F;"{;"|;!{;!|;!K;!|;"K;"r