Mathieu Functions

The routines described in this section compute the angular and radial Mathieu functions, and their characteristic values. Mathieu functions are the solutions of the following two differential equations:

\[\begin{split}d^2y/dv^2 + (a - 2q\cos 2v)y = 0 \\ d^2f/du^2 - (a - 2q\cosh 2u)f = 0\end{split}\]

The angular Mathieu functions \(ce_r(x,q), se_r(x,q)\) are the even and odd periodic solutions of the first equation, which is known as Mathieu’s equation. These exist only for the discrete sequence of characteristic values \(a=a_r(q)\) (even-periodic) and \(a=b_r(q)\) (odd-periodic).

The radial Mathieu functions \(Mc^{(j)}_{r}(z,q), Ms^{(j)}_{r}(z,q)\) are the solutions of the second equation, which is referred to as Mathieu’s modified equation. The radial Mathieu functions of the first, second, third and fourth kind are denoted by the parameter \(j\), which takes the value 1, 2, 3 or 4.

For more information on the Mathieu functions, see Abramowitz and Stegun, Chapter 20.

gsl_sf_mathieu_a(n, q)
gsl_sf_mathieu_b(n, q)

These routines compute the characteristic values \(a_n(q), b_n(q)\) of the Mathieu functions \(ce_n(q,x)\) and \(se_n(q,x)\), respectively.

Angular Mathieu Functions

gsl_sf_mathieu_ce(n, q, x)
gsl_sf_mathieu_se(n, q, x)

These routines compute the angular Mathieu functions \(ce_n(q,x)\) and \(se_n(q,x)\), respectively.

Radial Mathieu Functions

gsl_sf_mathieu_Mc(j, n, q, x)
gsl_sf_mathieu_Ms(j, n, q, x)

These routines compute the radial \(j\)-th kind Mathieu functions \(Mc_n^{(j)}(q,x)\) and \(Ms_n^{(j)}(q,x)\) of order \(n\).

The allowed values of \(j\) are \(1\) and \(2\). The functions for \(j = 3, 4\) can be computed as \(M_n^{(3)} = M_n^{(1)} + iM_n^{(2)}\) and \(M_n^{(4)} = M_n^{(1)} - iM_n^{(2)}\), where \(M_n^{(j)} = Mc_n^{(j)}\) or \(Ms_n^{(j)}\).