§30.4 Functions of the First Kind

Keywords:: spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.4

§30.4(i) Definitions
§30.4(ii) Elementary Properties
§30.4(iii) Power-Series Expansion
§30.4(iv) Orthogonality

§30.4(i) Definitions

Notes:: See Meixner and Schäfke (1954, §3.23).
Defines:: $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind
Keywords:: spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.4.i

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

30.4.1 $\int _{{-1}}^{1}\left(\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)\right)^{2}dx=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!},$

Symbols:: $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : 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

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

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

30.4.2 $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,0\right)=\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(x\right);$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer and $n\geq m$ : integer degree
Referenced by:: §30.4(iv)
Permalink:: http://dlmf.nist.gov/30.4.E2
Encodings:: TeX, pMML, png

compare §14.3(i).

§30.4(ii) Elementary Properties

Notes:: See Meixner and Schäfke (1954, §3.2).
Keywords:: spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.4.ii

30.4.3 $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(-x,\gamma^{2}\right)=(-1)^{{n-m}}\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right).$

Symbols:: $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.4.E3
Encodings:: TeX, pMML, png

$\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ has exactly $n-m$ zeros in the interval $-1<x<1$ .

§30.4(iii) Power-Series Expansion

Notes:: See Arscott (1964b, §8.2).
Permalink:: http://dlmf.nist.gov/30.4.iii

30.4.4 $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)=(1-x^{2})^{{\frac{1}{2}m}}\sum _{{k=0}}^{{\infty}}g_{k}x^{k},$ $-1\leq x\leq 1$ ,

Symbols:: $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : 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

where

30.4.5 $\alpha _{k}g_{{k+2}}+(\beta _{k}-\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right))g_{k}+\gamma _{k}g_{{k-2}}=0$

Symbols:: $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : 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

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(iv) Orthogonality

Notes:: See Meixner and Schäfke (1954, §3.23).
Keywords:: equiconvergent, spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.4.iv

30.4.6 $\int _{{-1}}^{1}\mathop{\mathsf{Ps}^{{m}}_{{k}}\/}\nolimits\!\left(x,\gamma^{2}\right)\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)dx=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!}\delta _{{k,n}}.$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : 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

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

30.4.7 $f(x)=\sum _{{n=m}}^{{\infty}}c_{n}\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right),$

Symbols:: $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : 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

where

30.4.8 $c_{n}=(n+\tfrac{1}{2})\frac{(n-m)!}{(n+m)!}\int _{{-1}}^{1}f(t)\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(t,\gamma^{2}\right)dt.$

Symbols:: $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $f(x)$ : function and $c_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/30.4.E8
Encodings:: TeX, pMML, png

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

30.4.9 $\lim _{{N\to\infty}}\int _{{-1}}^{1}\left|f(x)-\sum _{{n=m}}^{N}c_{n}\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)\right|^{2}dx=0.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : 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

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 $-1\leq x\leq 1$ .

§30.4 Functions of the First Kind

Contents

§30.4(i) Definitions

§30.4(ii) Elementary Properties

§30.4(iii) Power-Series Expansion

§30.4(iv) Orthogonality