# §30.4 Functions of the First Kind

## §30.4(i) Definitions

The eigenfunctions of (30.2.1) that correspond to the eigenvalues are denoted by , . They are normalized by the condition

the sign of being when is even, and the sign of being when is odd.

When is the prolate angular spheroidal wave function, and when is the oblate angular spheroidal wave function. If , reduces to the Ferrers function :

compare §14.3(i).

## §30.4(iii) Power-Series Expansion

where

with , , from (30.3.6), and , for even if is odd and for odd if is even. Normalization of the coefficients is effected by application of (30.4.1).

## §30.4(iv) Orthogonality

The expansion (30.4.7) converges in the norm of , that is,

It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for .