§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 ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$, $n=m,m+1,m+2,\mathrm{\dots }$. They are normalized by the condition

30.4.1		$${\int }_{-1}^1{\left({\mathsf{Ps}}_n^m(x,{\gamma }^2)\right)}^{2}dx=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!},$$
ⓘ Symbols: $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter A&S Ref: 21.7.11 Referenced by: §30.4(iii) Permalink: http://dlmf.nist.gov/30.4.E1 Encodings: TeX, pMML, png See also: Annotations for §30.4(i), §30.4 and Ch.30

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

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

30.4.2		$${\mathsf{Ps}}_n^m(x,0)={\mathsf{P}}_n^m\left(x\right);$$
ⓘ Symbols: : Ferrers function of the first kind, ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer and $n\ge m$: integer degree Referenced by: §30.4(iv) Permalink: http://dlmf.nist.gov/30.4.E2 Encodings: TeX, pMML, png See also: Annotations for §30.4(i), §30.4 and Ch.30

compare §14.3(i).

30.4.3		$${\mathsf{Ps}}_n^m(-x,{\gamma }^2)={(-1)}^{n-m}{\mathsf{Ps}}_n^m(x,{\gamma }^2).$$
ⓘ Symbols: ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter Permalink: http://dlmf.nist.gov/30.4.E3 Encodings: TeX, pMML, png See also: Annotations for §30.4(ii), §30.4 and Ch.30

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

30.4.4		$${\mathsf{Ps}}_n^m(x,{\gamma }^2)={(1-x^2)}^{\frac{1}{2}m}\sum _{k=0}^{\mathrm{\infty }}{g}_k{x}^k,$$
$-1\le x\le 1$,
ⓘ Symbols: ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter A&S Ref: 21.7.18 Permalink: http://dlmf.nist.gov/30.4.E4 Encodings: TeX, pMML, png See also: Annotations for §30.4(iii), §30.4 and Ch.30

where

30.4.5		$${\alpha }_k{g}_{k+2}+({\beta }_k-{\lambda }_n^m\left({\gamma }^2\right))g_k+{\gamma }_k{g}_{k-2}=0$$
ⓘ Symbols: ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, ${\alpha }_k$: coefficent, ${\beta }_k$: coefficent and ${\gamma }_k$: coefficent A&S Ref: 21.7.18 Permalink: http://dlmf.nist.gov/30.4.E5 Encodings: TeX, pMML, png See also: Annotations for §30.4(iii), §30.4 and Ch.30

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).

30.4.6		$${\int }_{-1}^1{\mathsf{Ps}}_k^m(x,{\gamma }^2){\mathsf{Ps}}_n^m(x,{\gamma }^2)dx=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!}{\delta }_{k,n}.$$
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter A&S Ref: 21.7.11 Permalink: http://dlmf.nist.gov/30.4.E6 Encodings: TeX, pMML, png See also: Annotations for §30.4(iv), §30.4 and Ch.30

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

30.4.7		$$f(x)=\sum _{n=m}^{\mathrm{\infty }}{c}_n{\mathsf{Ps}}_n^m(x,{\gamma }^2),$$
ⓘ Symbols: ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $f(x)$: function and $c_n$: coefficient Referenced by: §30.4(iv) Permalink: http://dlmf.nist.gov/30.4.E7 Encodings: TeX, pMML, png See also: Annotations for §30.4(iv), §30.4 and Ch.30

where

30.4.8		$$c_n=(n+\frac{1}{2})\frac{(n-m)!}{(n+m)!}{\int }_{-1}^{1}f(t){\mathsf{Ps}}_n^m(t,{\gamma }^2)dt.$$
ⓘ Defines: $c_n$: coefficient (locally) Symbols: $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter and $f(x)$: function Permalink: http://dlmf.nist.gov/30.4.E8 Encodings: TeX, pMML, png See also: Annotations for §30.4(iv), §30.4 and Ch.30

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

30.4.9		$$\underset{N\to \mathrm{\infty }}{lim}{\int }_{-1}^1{\left|f(x)-\sum _{n=m}^N{c}_n{\mathsf{Ps}}_n^m(x,{\gamma }^2)\right|}^{2}dx=0.$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $f(x)$: function and $c_n$: coefficient Permalink: http://dlmf.nist.gov/30.4.E9 Encodings: TeX, pMML, png See also: Annotations for §30.4(iv), §30.4 and Ch.30

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 $-1\le x\le 1$.