# §30.9 Asymptotic Approximations and Expansions

## §30.9(i) Prolate Spheroidal Wave Functions

As $\gamma^{2}\to+\infty$, with $q=2(n-m)+1$,

 30.9.1 $\lambda^{m}_{n}\left(\gamma^{2}\right)\sim-\gamma^{2}+\gamma q+\beta_{0}+\beta% _{1}\gamma^{-1}+\beta_{2}\gamma^{-2}+\cdots,$

where

 30.9.2 $\displaystyle 8\beta_{0}$ $\displaystyle=8m^{2}-q^{2}-5,$ $\displaystyle 2^{6}\beta_{1}$ $\displaystyle=-q^{3}-11q+32m^{2}q,$ $\displaystyle 2^{10}\beta_{2}$ $\displaystyle=-5(q^{4}+26q^{2}+21)+384m^{2}(q^{2}+1),$ $\displaystyle 2^{14}\beta_{3}$ $\displaystyle=-33q^{5}-1594q^{3}-5621q+128m^{2}(37q^{3}+167q)-2048m^{4}q.$ ⓘ Defines: $\beta_{n}$: coeffients (locally) Symbols: $m$: nonnegative integer, $n\geq m$: integer degree and $q$ Permalink: http://dlmf.nist.gov/30.9.E2 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §30.9(i), §30.9 and Ch.30
 30.9.3 $\displaystyle 2^{16}\beta_{4}$ $\displaystyle=-63q^{6}-4940q^{4}-43327q^{2}-22470+128m^{2}(115q^{4}+1310q^{2}+% 735)-24576m^{4}(q^{2}+1),$ $\displaystyle 2^{20}\beta_{5}$ $\displaystyle=-527q^{7}-61529q^{5}-10\;43961q^{3}-22\;41599q+32m^{2}(5739q^{5}% +1\;27550q^{3}+2\;98951q)-2048m^{4}(355q^{3}+1505q)+65536m^{6}q.$ ⓘ Symbols: $m$: nonnegative integer, $q$ and $\beta_{n}$: coeffients Permalink: http://dlmf.nist.gov/30.9.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §30.9(i), §30.9 and Ch.30

Further coefficients can be found with the Maple program SWF7; see §30.18(i).

For the eigenfunctions see Meixner and Schäfke (1954, §3.251) and Müller (1963).

For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). See also Miles (1975).

## §30.9(ii) Oblate Spheroidal Wave Functions

As $\gamma^{2}\to-\infty$, with $q=n+1$ if $n-m$ is even, or $q=n$ if $n-m$ is odd, we have

 30.9.4 $\lambda^{m}_{n}\left(\gamma^{2}\right)\sim 2q|\gamma|+c_{0}+c_{1}|\gamma|^{-1}% +c_{2}|\gamma|^{-2}+\cdots,$

where

 30.9.5 $\displaystyle 2c_{0}$ $\displaystyle=-q^{2}-1+m^{2},$ $\displaystyle 8c_{1}$ $\displaystyle=-q^{3}-q+m^{2}q,$ $\displaystyle 2^{6}c_{2}$ $\displaystyle=-5q^{4}-10q^{2}-1+2m^{2}(3q^{2}+1)-m^{4},$ $\displaystyle 2^{9}c_{3}$ $\displaystyle=-33q^{5}-114q^{3}-37q+2m^{2}(23q^{3}+25q)-13m^{4}q.$ ⓘ Symbols: $m$: nonnegative integer, $q$ and $c_{n}$: coeffients Permalink: http://dlmf.nist.gov/30.9.E5 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §30.9(ii), §30.9 and Ch.30
 30.9.6 $\displaystyle 2^{10}c_{4}$ $\displaystyle=-63q^{6}-340q^{4}-239q^{2}-14+10m^{2}(10q^{4}+23q^{2}+3)-3m^{4}(% 13q^{2}+6)+2m^{6},$ $\displaystyle 2^{13}c_{5}$ $\displaystyle=-527q^{7}-4139q^{5}-5221q^{3}-1009q+m^{2}(939q^{5}+3750q^{3}+159% 1q)-m^{4}(465q^{3}+635q)+53m^{6}q.$ ⓘ Symbols: $m$: nonnegative integer, $q$ and $c_{n}$: coeffients Permalink: http://dlmf.nist.gov/30.9.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §30.9(ii), §30.9 and Ch.30

Further coefficients can be found with the Maple program SWF8; see §30.18(i).

For the eigenfunctions see Meixner and Schäfke (1954, §3.252) and Müller (1962).

For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). See also Jorna and Springer (1971).

## §30.9(iii) Other Approximations and Expansions

The asymptotic behavior of $\lambda^{m}_{n}\left(\gamma^{2}\right)$ and $a^{m}_{n,k}(\gamma^{2})$ as $n\to\infty$ in descending powers of $2n+1$ is derived in Meixner (1944). The cases of large $m$, and of large $m$ and large $|\gamma|$, are studied in Abramowitz (1949). The asymptotic behavior of $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ and $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)$ as $x\to\pm 1$ is given in Erdélyi et al. (1955, p. 151). The behavior of $\lambda^{m}_{n}\left(\gamma^{2}\right)$ for complex $\gamma^{2}$ and large $|\lambda^{m}_{n}\left(\gamma^{2}\right)|$ is investigated in Hunter and Guerrieri (1982).