`scipy.special._basic:mathieu_even_coef`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_basic.py:1689

Signature

def mathieu_even_coef(m, q)

Summary

Fourier coefficients for even Mathieu and modified Mathieu functions.

Extended Summary

The Fourier series of the even solutions of the Mathieu differential equation are of the form

ce_{2 n} (z, q) = k = 0 \sum \infty A_{(2 n)}^{(2 k)} cos 2 k z

ce_{2 n + 1} (z, q) = k = 0 \sum \infty A_{(2 n + 1)}^{(2 k + 1)} cos (2 k + 1) z

This function returns the coefficients $A_{(2 n)}^{(2 k)}$ for even input m=2n, and the coefficients $A_{(2 n + 1)}^{(2 k + 1)}$ for odd input m=2n+1.

Parameters

m : int: Order of Mathieu functions. Must be non-negative.
q : float (>=0): Parameter of Mathieu functions. Must be non-negative.

Returns

Ak : ndarray: Even or odd Fourier coefficients, corresponding to even or odd m. The number of coefficients returned is determined by an empirical formula that depends on m and q ^[1].

Examples

We use the Fourier coefficients to construct an approximation of ``mathieu_cem(5, 14, x)``, the even Mathieu function of order `m = 5` and parameter `q = 14`.

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import mathieu_even_coef, mathieu_cem
m = 5
q = 14

✓

``a`` holds the Fourier coefficients. As noted above, the number of coefficients returned by ``mathieu_even_coef(m, q)`` is based on an empirical formula that depends on `m` and `q`. In this case, we get 29 coefficients.

a = mathieu_even_coef(m, q)
a.shape

✓

(29,)

Sum the Fourier cosine series on a grid of ``x`` values.

period = 180 if m % 2 == 0 else 360
x = np.linspace(0, period, 5000)             # x has shape (5000,)
k = np.arange(len(a)).reshape((-1, 1))       # k has shape (len(a), 1)
c = np.cos((2*k + m % 2) * (np.pi/180) * x)  # c has shape (len(a), 5000)
y = a @ c                                    # y has shape (5000,)

✓

Plot the approximation, along with the function computed directly by ``mathieu_cem(m, q, x)``.

plt.plot(x, y, 'k--', label="Fourier sum")

✗

[<matplotlib.lines.Line2D object at 0x7f3f591e97f0>]

ce, _dce = mathieu_cem(m, q, x)

✓

plt.plot(x, ce, alpha=0.35, linewidth=3.5, label="mathieu_cem")

✗

[<matplotlib.lines.Line2D object at 0x7f3f591e9940>]

plt.grid(True)

✓

plt.title(f'Mathieu Function $\\rm{{ce_{m}}}(x, {q})$')
plt.xlabel('x [degrees]')
plt.legend(shadow=True, loc='upper left', bbox_to_anchor=(1, 1))

✗

<matplotlib.legend.Legend object at 0x7f3f591e9550>

plt.tight_layout()
plt.show()

✓

Aliases

scipy.special.mathieu_even_coef