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 .
The eigenvalues are analytic functions of the real variable and satisfy
If is an even nonnegative integer, then the continued-fraction equation
where , , are defined by
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