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.
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gsl_sf_mathieu_a(n, q)
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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
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gsl_sf_mathieu_ce(n, q, x)
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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
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gsl_sf_mathieu_Mc(j, n, q, x)
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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 \(Ms_n^{(j)}\).