# §30.3 Eigenvalues

## §30.3(i) Definition

With , the spheroidal wave functions are solutions of Equation (30.2.1) which are bounded on , or equivalently, which are of the form where is an entire function of . These solutions exist only for eigenvalues , , of the parameter .

## §30.3(iii) Transcendental Equation

If is an even nonnegative integer, then the continued-fraction equation

30.3.5

where , , are defined by

30.3.6

has the solutions , . If is an odd positive integer, then Equation (30.3.5) has the solutions , . If or , the finite continued-fraction on the left-hand side of (30.3.5) equals 0; if its last denominator is or .

In equation (30.3.5) we can also use

## §30.3(iv) Power-Series Expansion

For values of see Meixner et al. (1980, p. 109).

30.3.9
30.3.11
30.3.12

Further coefficients can be found with the Maple program SWF9; see §30.18(i).