Angular Spheroidal Functions as a Function of Spheroidicity

This Demonstration shows how the angular spheroidal functions, , vary over the interval . For comparison we also show the corresponding Legendre functions, , to which the spheroidal ones reduce when . The controls allow to be varied: for (real ) we have the so-called prolate functions, while for (imaginary ) we have the oblate functions.

Snapshot 1: in the prolate limit (where the spheroid becomes an infinite cylinder), the angular spheroidal functions become more concentrated around the origin (), and less concentrated around the edges ()

Snapshot 2: the reverse holds in the oblate limit , in which the spheroid becomes a flat disk

Snapshot 3: in the oblate limit, both even and odd functions vanish at ; the even functions tend to the same form as the next higher odd function, with a sign change on one half of the interval (compare this picture, for , with Snapshot 2, where )