WOLFRAM|DEMONSTRATIONS PROJECT

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
.