30 Spheroidal Wave Functions Properties 30.7 Graphics 30.9 Asymptotic Approximations and Expansions

§30.8 Expansions in Series of Ferrers Functions

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Keywords:: expansions in series of Ferrers functions, spheroidal wave functions
Referenced by:: §30.16(i)
Permalink:: http://dlmf.nist.gov/30.8
See also:: Annotations for Ch.30

§30.8(i) Functions of the First Kind
§30.8(ii) Functions of the Second Kind

§30.8(i) Functions of the First Kind

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Keywords:: asymptotic behavior of coefficients, expansions in series of Ferrers functions, spheroidal wave functions
Notes:: See Arscott (1964b, §8.2) and Meixner and Schäfke (1954, §§3.542, 3.62).
Referenced by:: §30.8(ii)
Permalink:: http://dlmf.nist.gov/30.8.i
See also:: Annotations for §30.8 and Ch.30

30.8.1 $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)=\sum_{k=-R}^{\infty}(-1)^{k}a^{m}% _{n,k}(\gamma^{2})\mathsf{P}^{m}_{n+2k}\left(x\right),$

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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, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $R=\left\lfloor(n-m)/2\right\rfloor$ : limit and $a^{m}_{n,k}(\gamma^{2})$ : coefficients
A&S Ref:: 21.7.1 (in different form)
Referenced by:: §30.16(ii), §30.16(ii)
Permalink:: http://dlmf.nist.gov/30.8.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §30.8(i), §30.8 and Ch.30

where $\mathsf{P}^{m}_{n+2k}\left(x\right)$ is the Ferrers function of the first kind (§14.3(i)), $R=\left\lfloor\frac{1}{2}(n-m)\right\rfloor$ , and the coefficients $a^{m}_{n,k}(\gamma^{2})$ are given by

30.8.2 $a^{m}_{n,k}(\gamma^{2})=(-1)^{k}\left(n+2k+\tfrac{1}{2}\right)\frac{(n-m+2k)!}% {(n+m+2k)!}\*\int_{-1}^{1}\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)\mathsf{% P}^{m}_{n+2k}\left(x\right)\,\mathrm{d}x.$

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Defines:: $a^{m}_{n,k}(\gamma^{2})$ : coefficients (locally)
Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\,\mathrm{d}\NVar{x}$ : 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
Referenced by:: §30.11(i)
Permalink:: http://dlmf.nist.gov/30.8.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §30.8(i), §30.8 and Ch.30

Let

30.8.3

$A_{k}=-\gamma^{2}\frac{(n-m+2k-1)(n-m+2k)}{(2n+4k-3)(2n+4k-1)},$

$B_{k}=(n+2k)(n+2k+1)-2\gamma^{2}\frac{(n+2k)(n+2k+1)-1+m^{2}}{(2n+4k-1)(2n+4k+% 3)},$

$C_{k}=-\gamma^{2}\frac{(n+m+2k+1)(n+m+2k+2)}{(2n+4k+3)(2n+4k+5)}.$

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Defines:: $A_{k}$ (locally), $B_{k}$ (locally) and $C_{k}$ (locally)
Symbols:: : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.8.E3
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §30.8(i), §30.8 and Ch.30

Then the set of coefficients $a^{m}_{n,k}(\gamma^{2})$ , $k=-R,-R+1,-R+2,\dots$ is the solution of the difference equation

30.8.4 $A_{k}f_{k-1}+\left(B_{k}-\lambda^{m}_{n}\left(\gamma^{2}\right)\right)f_{k}+C_% {k}f_{k+1}=0,$

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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, $A_{k}$ , $B_{k}$ and $C_{k}$
Referenced by:: §30.16(ii), §30.8(ii)
Permalink:: http://dlmf.nist.gov/30.8.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §30.8(i), §30.8 and Ch.30

(note that $A_{-R}=0$ ) that satisfies the normalizing condition

30.8.5 $\sum_{k=-R}^{\infty}a_{n,k}^{m}(\gamma^{2})a_{n,k}^{-m}(\gamma^{2})\frac{1}{2n% +4k+1}=\frac{1}{2n+1},$

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Symbols:: : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $R=\left\lfloor(n-m)/2\right\rfloor$ : limit and $a^{m}_{n,k}(\gamma^{2})$ : coefficients
Referenced by:: §30.16(ii)
Permalink:: http://dlmf.nist.gov/30.8.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §30.8(i), §30.8 and Ch.30

with

30.8.6 $a_{n,k}^{-m}(\gamma^{2})=\frac{(n-m)!(n+m+2k)!}{(n+m)!(n-m+2k)!}a_{n,k}^{m}(% \gamma^{2}).$

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Symbols:: : factorial (as in ), : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter and $a^{m}_{n,k}(\gamma^{2})$ : coefficients
Referenced by:: §30.11(i)
Permalink:: http://dlmf.nist.gov/30.8.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §30.8(i), §30.8 and Ch.30

Also, as $k\to\infty$ ,

30.8.7 $\frac{k^{2}a^{m}_{n,k}(\gamma^{2})}{a^{m}_{n,k-1}(\gamma^{2})}=\frac{\gamma^{2% }}{16}+O\left(\frac{1}{k}\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter and $a^{m}_{n,k}(\gamma^{2})$ : coefficients
Permalink:: http://dlmf.nist.gov/30.8.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §30.8(i), §30.8 and Ch.30

and

30.8.8 $\frac{\lambda^{m}_{n}\left(\gamma^{2}\right)-B_{k}}{A_{k}}\frac{a^{m}_{n,k}(% \gamma^{2})}{a^{m}_{n,k-1}(\gamma^{2})}=1+O\left(\frac{1}{k^{4}}\right).$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\lambda^{\NVar{m}}_{\NVar{n}}\left(\NVar{\gamma^{2}}\right)$ : eigenvalues of the spheroidal differential equation, : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $a^{m}_{n,k}(\gamma^{2})$ : coefficients, $A_{k}$ and $B_{k}$
Permalink:: http://dlmf.nist.gov/30.8.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §30.8(i), §30.8 and Ch.30

§30.8(ii) Functions of the Second Kind

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Keywords:: expansions in series of Ferrers functions, spheroidal wave functions
Notes:: See Arscott (1964b, §8.5) and Meixner and Schäfke (1954, §3.62).
Referenced by:: §30.5
Permalink:: http://dlmf.nist.gov/30.8.ii
See also:: Annotations for §30.8 and Ch.30

30.8.9 $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)=\sum_{k=-\infty}^{-N-1}(-1)^{k}{a% ^{\prime}}^{m}_{n,k}(\gamma^{2})\mathsf{P}^{m}_{n+2k}\left(x\right)+\sum_{k=-N% }^{\infty}(-1)^{k}a^{m}_{n,k}(\gamma^{2})\mathsf{Q}^{m}_{n+2k}\left(x\right),$

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the second kind, : real variable, : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $a^{m}_{n,k}(\gamma^{2})$ : coefficients and : upper limit
A&S Ref:: 21.7.21 (in different form)
Permalink:: http://dlmf.nist.gov/30.8.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §30.8(ii), §30.8 and Ch.30

where $\mathsf{P}^{m}_{n}$ and $\mathsf{Q}^{m}_{n}$ are again the Ferrers functions and $N=\left\lfloor\frac{1}{2}(n+m)\right\rfloor$ . The coefficients $a^{m}_{n,k}(\gamma^{2})$ satisfy (30.8.4) for all when we set $a^{m}_{n,k}(\gamma^{2})=0$ for . For $k\geq-R$ they agree with the coefficients defined in §30.8(i). For $k=-N,-N+1,\dots,-R-1$ they are determined from (30.8.4) by forward recursion using $a^{m}_{n,-N-1}(\gamma^{2})=0$ . The set of coefficients ${a^{\prime}}^{m}_{n,k}(\gamma^{2})$ , $k=-N-1,-N-2,\dots$ , is the recessive solution of (30.8.4) as $k\to-\infty$ that is normalized by

30.8.10 $A_{-N-1}{a^{\prime}}^{m}_{n,-N-2}(\gamma^{2})+{\left(B_{-N-1}-\lambda^{m}_{n}% \left(\gamma^{2}\right)\right){a^{\prime}}^{m}_{n,-N-1}(\gamma^{2})}+C^{\prime% }a^{m}_{n,-N}(\gamma^{2})=0,$

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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, $a^{m}_{n,k}(\gamma^{2})$ : coefficients, $A_{k}$ , $B_{k}$ , : upper limit and $C^{\prime}$
Permalink:: http://dlmf.nist.gov/30.8.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §30.8(ii), §30.8 and Ch.30

with

30.8.11 $C^{\prime}=\begin{cases}\dfrac{\gamma^{2}}{4m^{2}-1},&\mbox{$n-m$ even},\\ -\dfrac{\gamma^{2}}{(2m-1)(2m-3)},&\mbox{$n-m$ odd}.\end{cases}$

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Defines:: $C^{\prime}$ (locally)
Symbols:: : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.8.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §30.8(ii), §30.8 and Ch.30

It should be noted that if the forward recursion (30.8.4) beginning with $f_{-N-1}=0$ , $f_{-N}=1$ leads to $f_{-R}=0$ , then $a^{m}_{n,k}(\gamma^{2})$ is undefined for and $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)$ does not exist.

30.7 Graphics 30.9 Asymptotic Approximations and Expansions

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§30.8 Expansions in Series of Ferrers Functions

Contents

§30.8(i) Functions of the First Kind

§30.8(ii) Functions of the Second Kind