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""" This module contains the Mathieu functions. |
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""" |
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from sympy.core.function import DefinedFunction, ArgumentIndexError |
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from sympy.functions.elementary.miscellaneous import sqrt |
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from sympy.functions.elementary.trigonometric import sin, cos |
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class MathieuBase(DefinedFunction): |
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""" |
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Abstract base class for Mathieu functions. |
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This class is meant to reduce code duplication. |
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""" |
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unbranched = True |
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def _eval_conjugate(self): |
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a, q, z = self.args |
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return self.func(a.conjugate(), q.conjugate(), z.conjugate()) |
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class mathieus(MathieuBase): |
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r""" |
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The Mathieu Sine function $S(a,q,z)$. |
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Explanation |
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=========== |
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This function is one solution of the Mathieu differential equation: |
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.. math :: |
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y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0 |
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The other solution is the Mathieu Cosine function. |
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Examples |
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======== |
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>>> from sympy import diff, mathieus |
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>>> from sympy.abc import a, q, z |
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>>> mathieus(a, q, z) |
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mathieus(a, q, z) |
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>>> mathieus(a, 0, z) |
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sin(sqrt(a)*z) |
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>>> diff(mathieus(a, q, z), z) |
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mathieusprime(a, q, z) |
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See Also |
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======== |
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mathieuc: Mathieu cosine function. |
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mathieusprime: Derivative of Mathieu sine function. |
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mathieucprime: Derivative of Mathieu cosine function. |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Mathieu_function |
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.. [2] https://dlmf.nist.gov/28 |
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.. [3] https://mathworld.wolfram.com/MathieuFunction.html |
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.. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuS/ |
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""" |
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def fdiff(self, argindex=1): |
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if argindex == 3: |
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a, q, z = self.args |
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return mathieusprime(a, q, z) |
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else: |
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raise ArgumentIndexError(self, argindex) |
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@classmethod |
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def eval(cls, a, q, z): |
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if q.is_Number and q.is_zero: |
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return sin(sqrt(a)*z) |
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if z.could_extract_minus_sign(): |
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return -cls(a, q, -z) |
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class mathieuc(MathieuBase): |
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r""" |
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The Mathieu Cosine function $C(a,q,z)$. |
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Explanation |
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=========== |
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This function is one solution of the Mathieu differential equation: |
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.. math :: |
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y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0 |
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The other solution is the Mathieu Sine function. |
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Examples |
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======== |
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>>> from sympy import diff, mathieuc |
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>>> from sympy.abc import a, q, z |
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>>> mathieuc(a, q, z) |
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mathieuc(a, q, z) |
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>>> mathieuc(a, 0, z) |
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cos(sqrt(a)*z) |
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>>> diff(mathieuc(a, q, z), z) |
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mathieucprime(a, q, z) |
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See Also |
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======== |
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mathieus: Mathieu sine function |
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mathieusprime: Derivative of Mathieu sine function |
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mathieucprime: Derivative of Mathieu cosine function |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Mathieu_function |
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.. [2] https://dlmf.nist.gov/28 |
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.. [3] https://mathworld.wolfram.com/MathieuFunction.html |
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.. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuC/ |
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""" |
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def fdiff(self, argindex=1): |
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if argindex == 3: |
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a, q, z = self.args |
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return mathieucprime(a, q, z) |
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else: |
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raise ArgumentIndexError(self, argindex) |
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@classmethod |
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def eval(cls, a, q, z): |
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if q.is_Number and q.is_zero: |
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return cos(sqrt(a)*z) |
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if z.could_extract_minus_sign(): |
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return cls(a, q, -z) |
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class mathieusprime(MathieuBase): |
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r""" |
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The derivative $S^{\prime}(a,q,z)$ of the Mathieu Sine function. |
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Explanation |
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=========== |
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This function is one solution of the Mathieu differential equation: |
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.. math :: |
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y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0 |
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The other solution is the Mathieu Cosine function. |
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Examples |
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======== |
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>>> from sympy import diff, mathieusprime |
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>>> from sympy.abc import a, q, z |
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>>> mathieusprime(a, q, z) |
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mathieusprime(a, q, z) |
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>>> mathieusprime(a, 0, z) |
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sqrt(a)*cos(sqrt(a)*z) |
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>>> diff(mathieusprime(a, q, z), z) |
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(-a + 2*q*cos(2*z))*mathieus(a, q, z) |
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See Also |
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======== |
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mathieus: Mathieu sine function |
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mathieuc: Mathieu cosine function |
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mathieucprime: Derivative of Mathieu cosine function |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Mathieu_function |
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.. [2] https://dlmf.nist.gov/28 |
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.. [3] https://mathworld.wolfram.com/MathieuFunction.html |
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.. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuSPrime/ |
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""" |
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def fdiff(self, argindex=1): |
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if argindex == 3: |
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a, q, z = self.args |
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return (2*q*cos(2*z) - a)*mathieus(a, q, z) |
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else: |
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raise ArgumentIndexError(self, argindex) |
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@classmethod |
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def eval(cls, a, q, z): |
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if q.is_Number and q.is_zero: |
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return sqrt(a)*cos(sqrt(a)*z) |
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if z.could_extract_minus_sign(): |
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return cls(a, q, -z) |
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class mathieucprime(MathieuBase): |
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r""" |
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The derivative $C^{\prime}(a,q,z)$ of the Mathieu Cosine function. |
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Explanation |
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=========== |
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This function is one solution of the Mathieu differential equation: |
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.. math :: |
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y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0 |
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The other solution is the Mathieu Sine function. |
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Examples |
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======== |
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>>> from sympy import diff, mathieucprime |
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>>> from sympy.abc import a, q, z |
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>>> mathieucprime(a, q, z) |
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mathieucprime(a, q, z) |
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>>> mathieucprime(a, 0, z) |
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-sqrt(a)*sin(sqrt(a)*z) |
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>>> diff(mathieucprime(a, q, z), z) |
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(-a + 2*q*cos(2*z))*mathieuc(a, q, z) |
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See Also |
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======== |
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mathieus: Mathieu sine function |
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mathieuc: Mathieu cosine function |
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mathieusprime: Derivative of Mathieu sine function |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Mathieu_function |
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.. [2] https://dlmf.nist.gov/28 |
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.. [3] https://mathworld.wolfram.com/MathieuFunction.html |
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.. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuCPrime/ |
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""" |
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def fdiff(self, argindex=1): |
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if argindex == 3: |
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a, q, z = self.args |
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return (2*q*cos(2*z) - a)*mathieuc(a, q, z) |
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else: |
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raise ArgumentIndexError(self, argindex) |
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@classmethod |
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def eval(cls, a, q, z): |
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if q.is_Number and q.is_zero: |
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return -sqrt(a)*sin(sqrt(a)*z) |
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if z.could_extract_minus_sign(): |
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return -cls(a, q, -z) |
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