§30.4 Functions of the First Kind

The eigenfunctions of (30.2.1) that correspond to the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are denoted by ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, $n=m,m+1,m+2,\mathrm{\dots }$. They are normalized by the condition

the sign of ${\mathrm{𝖯𝗌}}_n^m(0,{\gamma }^2)$ being ${(-1)}^{(n+m)/2}$ when $n-m$ is even, and the sign of ${d{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)/dx|}_{x=0}$ being ${(-1)}^{(n+m-1)/2}$ when $n-m$ is odd.

When ${\gamma }^2>0$ ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ is the prolate angular spheroidal wave function, and when ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ is the oblate angular spheroidal wave function. If $\gamma =0$, ${\mathrm{𝖯𝗌}}_n^m(x,0)$ reduces to the Ferrers function ${𝖯}_n^m\left(x\right)$:

compare §14.3(i).

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ has exactly $n-m$ zeros in the interval .

where

with ${\alpha }_k$, ${\beta }_k$, ${\gamma }_k$ from (30.3.6), and $g_{-1}=g_{-2}=0$, $g_k=0$ for even $k$ if $n-m$ is odd and $g_k=0$ for odd $k$ if $n-m$ is even. Normalization of the coefficients $g_k$ is effected by application of (30.4.1).

If $f(x)$ is mean-square integrable on $[-1,1]$, then formally

where

The expansion (30.4.7) converges in the norm of $L^2(-1,1)$, that is,

It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for $-1\le x\le 1$.