+ iRt^RIHtHt^RIHt^RIHtHt!RR]4t !RR] 4t !RR ] 4t !R R ] 4t !R R ] 4t R#)z,This module contains the Mathieu functions. )DefinedFunctionArgumentIndexError)sqrt)sincosc.a]tRt^ toRtRtRtRtVtR#) MathieuBasez^ Abstract base class for Mathieu functions. This class is meant to reduce code duplication. TcVPwrpVPVP4VP4VP44#N)argsfunc conjugate)selfaqzs& g/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/functions/special/mathieu_functions.py_eval_conjugateMathieuBase._eval_conjugates4))ayy q{{}EEN) __name__ __module__ __qualname____firstlineno____doc__ unbranchedr__static_attributes____classdictcell__ __classdict__s@rrr sJFFrrc>a]tRt^toRtRRlt]R4tRtVt R#)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/ c\V^8XdVPwr#p\W#V4#\W4h)r mathieusprimerrargindexrrrs&& rfdiffmathieus.fdiffF. q=iiGA! q) )$T4 4rcVP'd.VP'd\\V4V,4#VP 4'd V!WV)4)#R#r ) is_Numberis_zerorrcould_extract_minus_signclsrrrs&&&&reval mathieus.evalMsI ;;;1999tAwqy> ! % % ' 'qbM> ! (rrN rrrrrr) classmethodr2rrrs@rr"r"$+Z5""rr"c>a]tRt^VtoRtRRlt]R4tRtVt R#)mathieucao 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\V^8XdVPwr#p\W#V4#\W4hr$)r mathieucprimerr's&& rr)mathieuc.fdiffr+rcVP'd.VP'd\\V4V,4#VP 4'd V!WV)4#R#r )r-r.rrr/r0s&&&&rr2 mathieuc.evalsG ;;;1999tAwqy> ! % % ' 'qaR=  (rrNr4r6rs@rr:r:V$+Z5!!rr:c>a]tRt^toRtRRlt]R4tRtVt R#)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/ cV^8XdHVPwr#p^V,\^V,4,V, \W#V4,#\W4hr$)r rr"rr's&& rr)mathieusprime.fdiffH q=iiGA!aCAaCL1$hqQ&77 7$T4 4rcVP'd>VP'd,\V4\\V4V,4,#VP 4'd V!WV)4#R#r )r-r.rrr/r0s&&&&rr2mathieusprime.evalsP ;;;199973tAwqy>) ) % % ' 'qaR=  (rrNr4r6rs@rr&r&r@rr&c>a]tRt^toRtRRlt]R4tRtVt 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/ cV^8XdHVPwr#p^V,\^V,4,V, \W#V4,#\W4hr$)r rr:rr's&& rr)mathieucprime.fdiffrDrcVP'd?VP'd-\V4)\\V4V,4,#VP 4'd V!WV)4)#R#r )r-r.rrr/r0s&&&&rr2mathieucprime.evalsT ;;;1999G8CQ N* * % % ' 'qbM> ! (rrNr4r6rs@rr<r<r8rr<N)rsympy.core.functionrr(sympy.functions.elementary.miscellaneousr(sympy.functions.elementary.trigonometricrrrr"r:r&r<rrrrOsWD9= F/ F;"{;"|;!{;!|;!K;!|;"K;"r