Closed Form Solutions for Spheroidal Functions

function

SpheroidalPS

SpheroidalQS

SpheroidalS1

SpheroidalS2

n

1

For general parameter values, the spheroidal functions

PS

n,m

(γ,z)

,

QS

n,m

(γ,z)

,

(1)

S

n,m

(γ,z)

, and

(2)

S

n,m

(γ,z)

are difficult to compute and are calculated via infinite series representations involving simpler special functions. However, for the special set of parameter values

m=1

,

n=1,2,…

, and

γ=nπ/2

, simple closed-form solutions can be found involving trigonometric functions. These provide a useful check on the accuracy of numerical solutions. In this Demonstration we illustrate these solutions by showing both the closed-form expression and plotting over the standard domain of

z

.