`scipy.special._basic:mathieu_odd_coef`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_basic.py:1798

Signature

def mathieu_odd_coef(m, q)

Summary

Fourier coefficients for odd Mathieu and modified Mathieu functions.

Extended Summary

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

se_{2 n + 1} (z, q) = k = 0 \sum \infty B_{(2 n + 1)}^{(2 k + 1)} sin (2 k + 1) z

se_{2 n + 2} (z, q) = k = 0 \sum \infty B_{(2 n + 2)}^{(2 k + 2)} sin (2 k + 2) z

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

Bk : 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_sem(5, 11, x)``, the odd Mathieu function of order `m = 5` and parameter `q = 11`.

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import mathieu_odd_coef, mathieu_sem
m = 5
q = 11

✓

``b`` holds the Fourier coefficients. As noted above, the number of coefficients returned by ``mathieu_odd_coef(m, q)`` is based on an empirical formula that depends on `m` and `q`. In this case, we get 28 coefficients.

b = mathieu_odd_coef(m, q)
b.shape

✓

(28,)

Sum the Fourier sine 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(1, len(b) + 1).reshape((-1, 1))  # k has shape (len(b), 1)
c = np.sin((2*k - m % 2) * (np.pi/180) * x)    # c has shape (len(b), 5000)
y = b @ c                                      # y has shape (5000,)

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

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

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[<matplotlib.lines.Line2D object at 0x7f3f5917e900>]

se, _sce = mathieu_sem(m, q, x)

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plt.plot(x, se, alpha=0.35, linewidth=3.5, label="mathieu_sem")

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[<matplotlib.lines.Line2D object at 0x7f3f5917e660>]

plt.grid(True)

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

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<matplotlib.legend.Legend object at 0x7f3f5917ef90>

plt.tight_layout()
plt.show()

✓

Aliases

scipy.special.mathieu_odd_coef