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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`.

• Mathieu Function Workspace: | ||

• Mathieu Function Characteristic Values: | ||

• Angular Mathieu Functions: | ||

• Radial Mathieu Functions: |