30 Spheroidal Wave Functions Properties 30.3 Eigenvalues 30.5 Functions of the Second Kind

§30.4 Functions of the First Kind

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Keywords:: of the first kind, spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.4
See also:: Annotations for Ch.30

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

§30.4(i) Definitions

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Defines:: $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind
Keywords:: definitions, oblate angular, prolate angular, spheroidal wave functions
Notes:: See Meixner and Schäfke (1954, §3.23).
Permalink:: http://dlmf.nist.gov/30.4.i
See also:: Annotations for §30.4 and Ch.30

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

30.4.1 $\int_{-1}^{1}\left(\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)\right)^{2}\,% \mathrm{d}x=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!},$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\int$ : integral, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, : real variable, : 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
See also:: Annotations for §30.4(i), §30.4 and Ch.30

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

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

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

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, : real variable, : 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
See also:: Annotations for §30.4(i), §30.4 and Ch.30

compare §14.3(i).

§30.4(ii) Elementary Properties

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Keywords:: elementary properties, spheroidal wave functions, zeros
Notes:: See Meixner and Schäfke (1954, §3.2).
Permalink:: http://dlmf.nist.gov/30.4.ii
See also:: Annotations for §30.4 and Ch.30

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

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

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

§30.4(iii) Power-Series Expansion

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Notes:: See Arscott (1964b, §8.2).
Permalink:: http://dlmf.nist.gov/30.4.iii
See also:: Annotations for §30.4 and Ch.30

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

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Symbols:: $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, : real variable, : 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
See also:: Annotations for §30.4(iii), §30.4 and Ch.30

where

30.4.5 $\alpha_{k}g_{k+2}+(\beta_{k}-\lambda^{m}_{n}\left(\gamma^{2}\right))g_{k}+% \gamma_{k}g_{k-2}=0$

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Symbols:: $\lambda^{\NVar{m}}_{\NVar{n}}\left(\NVar{\gamma^{2}}\right)$ : eigenvalues of the spheroidal differential equation, : 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
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 if is odd and $g_{k}=0$ for odd if is even. Normalization of the coefficients $g_{k}$ is effected by application of (30.4.1).

§30.4(iv) Orthogonality

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Keywords:: equiconvergent, orthogonality, spheroidal wave functions
Notes:: See Meixner and Schäfke (1954, §3.23).
Permalink:: http://dlmf.nist.gov/30.4.iv
See also:: Annotations for §30.4 and Ch.30

30.4.6 $\int_{-1}^{1}\mathsf{Ps}^{m}_{k}\left(x,\gamma^{2}\right)\mathsf{Ps}^{m}_{n}% \left(x,\gamma^{2}\right)\,\mathrm{d}x=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!}% \delta_{k,n}.$

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\int$ : integral, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, : real variable, : 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
See also:: Annotations for §30.4(iv), §30.4 and Ch.30

If is mean-square integrable on , then formally

30.4.7 $f(x)=\sum_{n=m}^{\infty}c_{n}\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),$

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Symbols:: $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, : real variable, : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, : 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 $c_{n}=(n+\tfrac{1}{2})\frac{(n-m)!}{(n+m)!}\int_{-1}^{1}f(t)\mathsf{Ps}^{m}_{n% }\left(t,\gamma^{2}\right)\,\mathrm{d}t.$

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Defines:: $c_{n}$ : coefficient (locally)
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\int$ : integral, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter and : 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 $\lim_{N\to\infty}\int_{-1}^{1}{\left|f(x)-\sum_{n=m}^{N}c_{n}\mathsf{Ps}^{m}_{% n}\left(x,\gamma^{2}\right)\right|}^{2}\,\mathrm{d}x=0.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, : real variable, : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, : 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 $-1\leq x\leq 1$ .

30.3 Eigenvalues 30.5 Functions of the Second Kind

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§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