# §30.9 Asymptotic Approximations and Expansions

## §30.9(i) Prolate Spheroidal Wave Functions

30.9.3

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 , with if is even, or if is odd, we have

where

30.9.5
30.9.6

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 and as in descending powers of is derived in Meixner (1944). The cases of large , and of large and large , are studied in Abramowitz (1949). The asymptotic behavior of and as is given in Erdélyi et al. (1955, p. 151). The behavior of for complex and large is investigated in Hunter and Guerrieri (1982).