ó Ąź™\c@ sÚdZddlmZmZddlmZddlmZmZddl m Z ddl m Z m Z defd„ƒYZd efd „ƒYZd efd „ƒYZd efd„ƒYZdefd„ƒYZdS(s- This module contains the Mathieu functions. i˙˙˙˙(tprint_functiontdivision(tS(tFunctiontArgumentIndexError(tsqrt(tsintcost MathieuBasecB seZdZeZd„ZRS(si Abstract base class for Mathieu functions. This class is meant to reduce code duplication. cC s7|j\}}}|j|jƒ|jƒ|jƒƒS(N(targstfunct conjugate(tselftatqtz((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyt_eval_conjugates(t__name__t __module__t__doc__tTruet unbranchedR(((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyR stmathieuscB s)eZdZdd„Zed„ƒZRS(sž The Mathieu Sine function `S(a,q,z)`. 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] http://dlmf.nist.gov/28 .. [3] http://mathworld.wolfram.com/MathieuBase.html .. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuS/ icC sA|dkr.|j\}}}t|||ƒSt||ƒ‚dS(Ni(R t mathieusprimeR(R targindexR RR((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pytfdiffCs cC sN|jr,|tjkr,tt|ƒ|ƒS|jƒrJ|||| ƒ SdS(N(t is_NumberRtZeroRRtcould_extract_minus_sign(tclsR RR((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pytevalJs (RRRRt classmethodR(((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyRs' tmathieuccB s)eZdZdd„Zed„ƒZRS(sš The Mathieu Cosine function `C(a,q,z)`. 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] http://dlmf.nist.gov/28 .. [3] http://mathworld.wolfram.com/MathieuBase.html .. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuC/ icC sA|dkr.|j\}}}t|||ƒSt||ƒ‚dS(Ni(R t mathieucprimeR(R RR RR((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyR|s cC sM|jr,|tjkr,tt|ƒ|ƒS|jƒrI|||| ƒSdS(N(RRRRRR(RR RR((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyRƒs (RRRRRR(((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyR Ss' RcB s)eZdZdd„Zed„ƒZRS(sř The derivative `S^{\prime}(a,q,z)` of the Mathieu Sine function. 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] http://dlmf.nist.gov/28 .. [3] http://mathworld.wolfram.com/MathieuBase.html .. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuSPrime/ icC s[|dkrH|j\}}}d|td|ƒ|t|||ƒSt||ƒ‚dS(Nii(R RRR(R RR RR((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyRľs *cC sW|jr6|tjkr6t|ƒtt|ƒ|ƒS|jƒrS|||| ƒSdS(N(RRRRRR(RR RR((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyRźs (RRRRRR(((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyRŒs' R!cB s)eZdZdd„Zed„ƒZRS(s÷ The derivative `C^{\prime}(a,q,z)` of the Mathieu Cosine function. 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] http://dlmf.nist.gov/28 .. [3] http://mathworld.wolfram.com/MathieuBase.html .. [4] http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuCPrime/ icC s[|dkrH|j\}}}d|td|ƒ|t|||ƒSt||ƒ‚dS(Nii(R RR R(R RR RR((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyRîs *cC sY|jr7|tjkr7t|ƒ tt|ƒ|ƒS|jƒrU|||| ƒ SdS(N(RRRRRR(RR RR((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyRős (RRRRRR(((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyR!Ĺs' N(Rt __future__RRt sympy.coreRtsympy.core.functionRRt(sympy.functions.elementary.miscellaneousRt(sympy.functions.elementary.trigonometricRRRRR RR!(((sHlib/python2.7/site-packages/sympy/functions/special/mathieu_functions.pyts999