\hSrSSKJrJr SSKJr SSKJrJr "SS\5r "SS\ 5r "S S \ 5r "S S \ 5r "S S\ 5r g)z,This module contains the Mathieu functions. )DefinedFunctionArgumentIndexError)sqrt)sincosc"\rSrSrSrSrSrSrg) MathieuBase z^ Abstract base class for Mathieu functions. This class is meant to reduce code duplication. TcURupnURUR5UR5UR55$N)argsfunc conjugate)selfaqzs a/var/www/auris/envauris/lib/python3.13/site-packages/sympy/functions/special/mathieu_functions.py_eval_conjugateMathieuBase._eval_conjugates4))ayy q{{}EEN)__name__ __module__ __qualname____firstlineno____doc__ unbranchedr__static_attributes__rrrr r sJFrr c2\rSrSrSrSSjr\S5rSrg)mathieusat 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/ cZUS:XaURup#n[X#U5$[X5eN)r mathieusprimerrargindexrrrs rfdiffmathieus.fdiffF. q=iiGA! q) )$T4 4rcUR(a(UR(a[[U5U-5$UR 5(a U"XU*5*$gr ) is_Numberis_zerorrcould_extract_minus_signclsrrrs reval mathieus.evalMsE ;;199tAwqy> ! % % ' 'qbM> ! (rrN rrrrrr) classmethodr2rrrrr!r!!+Z5""rr!c2\rSrSrSrSSjr\S5rSrg)mathieucVao 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/ cZUS:XaURup#n[X#U5$[X5er$)r mathieucprimerr's rr)mathieuc.fdiffr+rcUR(a(UR(a[[U5U-5$UR 5(a U"XU*5$gr )r-r.rrr/r0s rr2 mathieuc.evalsC ;;199tAwqy> ! % % ' 'qaR=  (rrNr4r6rrrr:r:V!+Z5!!rr:c2\rSrSrSrSSjr\S5rSrg)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/ cUS:Xa3URup#nSU-[SU-5-U- [X#U5-$[X5eNr%)r rr!rr's rr)mathieusprime.fdiffH q=iiGA!aCAaCL1$hqQ&77 7$T4 4rcUR(a4UR(a#[U5[[U5U-5-$UR 5(a U"XU*5$gr )r-r.rrr/r0s rr2mathieusprime.evalsL ;;19973tAwqy>) ) % % ' 'qaR=  (rrNr4r6rrrr&r&rArr&c2\rSrSrSrSSjr\S5rSrg)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/ cUS:Xa3URup#nSU-[SU-5-U- [X#U5-$[X5erE)r rr:rr's rr)mathieucprime.fdiffrHrcUR(a5UR(a$[U5*[[U5U-5-$UR 5(a U"XU*5*$gr )r-r.rrr/r0s rr2mathieucprime.evalsP ;;199G8CQ N* * % % ' 'qbM> ! 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