scipy.special.mathieu_sem#

scipy.special.mathieu_sem(m, q, x, out=None) = <wrapped_ufunc 'mathieu_sem'>#

Odd Mathieu function and its derivative.

Returns the odd Mathieu function, se_m(x, q), of order m and parameter q evaluated at x (given in degrees). Also returns the derivative with respect to x of se_m(x, q).

Parameters:

marray_like: Order of the function
qarray_like: Parameter of the function
xarray_like: Argument of the function, given in degrees, not radians.
outtuple of ndarray, optional: Optional output arrays for the function results

Returns:

yscalar or ndarray: Value of the function
ypscalar or ndarray: Value of the derivative vs x

See also

mathieu_a, mathieu_b, mathieu_cem

Notes

Odd Mathieu functions are the solutions to Mathieu’s differential equation

\[\frac{d^2y}{dx^2} + (b_m - 2q \cos(2x))y = 0\]

for which the characteristic number \(b_m\) (calculated with mathieu_b) results in an odd, periodic solution \(y(x)\) with period 180 degrees (for even \(m\)) or 360 degrees (for odd \(m\)).

Array API Standard Support

mathieu_sem has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	⛔
Dask	⛔	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. mathieu_sem does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[1]

‘Mathieu function’. Wikipedia. https://en.wikipedia.org/wiki/Mathieu_function

[2]

Stuart Brorson, A New Implementation of the Mathieu Functions for SciPy. brorson/ScipyMathieuPaper

Examples

Plot odd Mathieu functions of orders 2 and 4.

>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> m = np.asarray([2, 4])
>>> q = 50
>>> x = np.linspace(-180, 180, 300)[:, np.newaxis]
>>> y, _ = special.mathieu_sem(m, q, x)
>>> plt.plot(x, y)
>>> plt.xlabel('x (degrees)')
>>> plt.ylabel('y')
>>> plt.legend(('m = 2', 'm = 4'))

Because the orders 2 and 4 are even, the period of each function is 180 degrees.

../../_images/scipy-special-mathieu_sem-1.png