Spaces:
Running
Running
| """ This module contains the Mathieu functions. | |
| """ | |
| from sympy.core.function import Function, ArgumentIndexError | |
| from sympy.functions.elementary.miscellaneous import sqrt | |
| from sympy.functions.elementary.trigonometric import sin, cos | |
| class MathieuBase(Function): | |
| """ | |
| Abstract base class for Mathieu functions. | |
| This class is meant to reduce code duplication. | |
| """ | |
| unbranched = True | |
| def _eval_conjugate(self): | |
| a, q, z = self.args | |
| return self.func(a.conjugate(), q.conjugate(), z.conjugate()) | |
| class mathieus(MathieuBase): | |
| r""" | |
| 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/ | |
| """ | |
| def fdiff(self, argindex=1): | |
| if argindex == 3: | |
| a, q, z = self.args | |
| return mathieusprime(a, q, z) | |
| else: | |
| raise ArgumentIndexError(self, argindex) | |
| def eval(cls, a, q, z): | |
| if q.is_Number and q.is_zero: | |
| return sin(sqrt(a)*z) | |
| # Try to pull out factors of -1 | |
| if z.could_extract_minus_sign(): | |
| return -cls(a, q, -z) | |
| class mathieuc(MathieuBase): | |
| r""" | |
| 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/ | |
| """ | |
| def fdiff(self, argindex=1): | |
| if argindex == 3: | |
| a, q, z = self.args | |
| return mathieucprime(a, q, z) | |
| else: | |
| raise ArgumentIndexError(self, argindex) | |
| def eval(cls, a, q, z): | |
| if q.is_Number and q.is_zero: | |
| return cos(sqrt(a)*z) | |
| # Try to pull out factors of -1 | |
| if z.could_extract_minus_sign(): | |
| return cls(a, q, -z) | |
| class mathieusprime(MathieuBase): | |
| r""" | |
| 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/ | |
| """ | |
| def fdiff(self, argindex=1): | |
| if argindex == 3: | |
| a, q, z = self.args | |
| return (2*q*cos(2*z) - a)*mathieus(a, q, z) | |
| else: | |
| raise ArgumentIndexError(self, argindex) | |
| def eval(cls, a, q, z): | |
| if q.is_Number and q.is_zero: | |
| return sqrt(a)*cos(sqrt(a)*z) | |
| # Try to pull out factors of -1 | |
| if z.could_extract_minus_sign(): | |
| return cls(a, q, -z) | |
| class mathieucprime(MathieuBase): | |
| r""" | |
| 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/ | |
| """ | |
| def fdiff(self, argindex=1): | |
| if argindex == 3: | |
| a, q, z = self.args | |
| return (2*q*cos(2*z) - a)*mathieuc(a, q, z) | |
| else: | |
| raise ArgumentIndexError(self, argindex) | |
| def eval(cls, a, q, z): | |
| if q.is_Number and q.is_zero: | |
| return -sqrt(a)*sin(sqrt(a)*z) | |
| # Try to pull out factors of -1 | |
| if z.could_extract_minus_sign(): | |
| return -cls(a, q, -z) | |