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:
d^2y/dv^2 + (a - 2q\cos 2v)y = 0
d^2f/du^2 - (a - 2q\cosh 2u)f = 0
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. These functions are defined in the header file gsl_sf_mathieu.h.