SciPy

scipy.special.mathieu_even_coef

scipy.special.mathieu_even_coef(m, q)[source]

Fourier coefficients for even Mathieu and modified Mathieu functions.

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

\[\mathrm{ce}_{2n}(z, q) = \sum_{k=0}^{\infty} A_{(2n)}^{(2k)} \cos 2kz\]
\[\mathrm{ce}_{2n+1}(z, q) = \sum_{k=0}^{\infty} A_{(2n+1)}^{(2k+1)} \cos (2k+1)z\]

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

References

[R322]Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996. http://jin.ece.illinois.edu/specfunc.html
[R323]NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/28.4#i