§30.9 Asymptotic Approximations and Expansions

Keywords:: spheroidal differential equation
Referenced by:: §30.16(i), §30.16(ii)
Permalink:: http://dlmf.nist.gov/30.9

§30.9(i) Prolate Spheroidal Wave Functions
§30.9(ii) Oblate Spheroidal Wave Functions
§30.9(iii) Other Approximations and Expansions

§30.9(i) Prolate Spheroidal Wave Functions

Notes:: See Meixner and Schäfke (1954, §3.251) and Müller (1963).
Keywords:: spheroidal wave functions
Referenced by:: §2.8(iii), §2.8(vi)
Permalink:: http://dlmf.nist.gov/30.9.i

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

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

Symbols:: $\sim$ : asymptotic equality, $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $q$ and $\beta _{n}$ : coeffients
A&S Ref:: 21.7.6
Referenced by:: §30.18(i)
Permalink:: http://dlmf.nist.gov/30.9.E1
Encodings:: TeX, pMML, png

where

30.9.2

$8\beta _{0}=8m^{2}-q^{2}-5,$

$2^{6}\beta _{1}=-q^{3}-11q+32m^{2}q,$

$2^{{10}}\beta _{2}=-5(q^{4}+26q^{2}+21)+384m^{2}(q^{2}+1),$

$2^{{14}}\beta _{3}=-33q^{5}-1594q^{3}-5621q+128m^{2}(37q^{3}+167q)-2048m^{4}q.$

Symbols:: $m$ : nonnegative integer, $n\geq m$ : integer degree, $q$ and $\beta _{n}$ : coeffients
Permalink:: http://dlmf.nist.gov/30.9.E2
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

30.9.3

$2^{{16}}\beta _{4}=-63q^{6}-4940q^{4}-43327q^{2}-22470+128m^{2}(115q^{4}+1310q^{2}+735)-24576m^{4}(q^{2}+1),$

$2^{{20}}\beta _{5}=-527q^{7}-61529q^{5}-10\; 43961q^{3}-22\; 41599q+32m^{2}(5739q^{5}+1\; 27550q^{3}+2\; 98951q)-2048m^{4}(355q^{3}+1505q)+65536m^{6}q.$

Symbols:: $m$ : nonnegative integer, $q$ and $\beta _{n}$ : coeffients
Permalink:: http://dlmf.nist.gov/30.9.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

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

Notes:: See Meixner and Schäfke (1954, §3.252) and Müller (1962).
Referenced by:: §2.8(iii), §2.8(vi)
Permalink:: http://dlmf.nist.gov/30.9.ii

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

Symbols:: $\sim$ : asymptotic equality, $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $q$ and $c_{n}$ : coeffients
A&S Ref:: 21.8.2
Referenced by:: §30.18(i)
Permalink:: http://dlmf.nist.gov/30.9.E4
Encodings:: TeX, pMML, png

where

30.9.5

$2c_{0}=-q^{2}-1+m^{2},$

$8c_{1}=-q^{3}-q+m^{2}q,$

$2^{6}c_{2}=-5q^{4}-10q^{2}-1+2m^{2}(3q^{2}+1)-m^{4},$

$2^{9}c_{3}=-33q^{5}-114q^{3}-37q+2m^{2}(23q^{3}+25q)-13m^{4}q.$

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

30.9.6

$2^{{10}}c_{4}=-63q^{6}-340q^{4}-239q^{2}-14+10m^{2}(10q^{4}+23q^{2}+3)-3m^{4}(13q^{2}+6)+2m^{6},$

$2^{{13}}c_{5}=-527q^{7}-4139q^{5}-5221q^{3}-1009q+m^{2}(939q^{5}+3750q^{3}+1591q)-m^{4}(465q^{3}+635q)+53m^{6}q.$

Symbols:: $m$ : nonnegative integer, $q$ and $c_{n}$ : coeffients
Permalink:: http://dlmf.nist.gov/30.9.E6
Encodings:: TeX, TeX, pMML, pMML, png, png

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

Keywords:: spheroidal differential equation, spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.9.iii

The asymptotic behavior of $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ 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 $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ and $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ as $x\to\pm 1$ is given in Erdélyi et al. (1955, p. 151). The behavior of $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ for complex $\gamma^{2}$ and large $|\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)|$ is investigated in Hunter and Guerrieri (1982).

§30.9 Asymptotic Approximations and Expansions

Contents

§30.9(i) Prolate Spheroidal Wave Functions

§30.9(ii) Oblate Spheroidal Wave Functions

§30.9(iii) Other Approximations and Expansions