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§30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

As γ2+, with q=2(n-m)+1,

30.9.1 λnm(γ2)-γ2+γq+β0+β1γ-1+β2γ-2+,

where

30.9.2 8β0 =8m2-q2-5,
26β1 =-q3-11q+32m2q,
210β2 =-5(q4+26q2+21)+384m2(q2+1),
214β3 =-33q5-1594q3-5621q+128m2(37q3+167q)-2048m4q.
30.9.3 216β4 =-63q6-4940q4-43327q2-22470+128m2(115q4+1310q2+735)-24576m4(q2+1),
220β5 =-527q7-61529q5-10 43961q3-22 41599q+32m2(5739q5+1 27550q3+2 98951q)-2048m4(355q3+1505q)+65536m6q.

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 γ2-, with q=n+1 if n-m is even, or q=n if n-m is odd, we have

30.9.4 λnm(γ2)2q|γ|+c0+c1|γ|-1+c2|γ|-2+,

where

30.9.5 2c0 =-q2-1+m2,
8c1 =-q3-q+m2q,
26c2 =-5q4-10q2-1+2m2(3q2+1)-m4,
29c3 =-33q5-114q3-37q+2m2(23q3+25q)-13m4q.
30.9.6 210c4 =-63q6-340q4-239q2-14+10m2(10q4+23q2+3)-3m4(13q2+6)+2m6,
213c5 =-527q7-4139q5-5221q3-1009q+m2(939q5+3750q3+1591q)-m4(465q3+635q)+53m6q.

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 λnm(γ2) and an,km(γ2) as n in descending powers of 2n+1 is derived in Meixner (1944). The cases of large m, and of large m and large |γ|, are studied in Abramowitz (1949). The asymptotic behavior of Psnm(x,γ2) and Qsnm(x,γ2) as x±1 is given in Erdélyi et al. (1955, p. 151). The behavior of λnm(γ2) for complex γ2 and large |λnm(γ2)| is investigated in Hunter and Guerrieri (1982).