BUILT-IN MATHEMATICA SYMBOL

SpheroidalPS

gives the angular spheroidal function

of the first kind.

The angular spheroidal functions satisfy the differential equation with the spheroidal eigenvalue given by SpheroidalEigenvalue.

SpheroidalPS gives spheroidal functions of type . The types are specified as for LegendreP.

For certain special arguments, SpheroidalPS automatically evaluates to exact values.

Evaluate numerically:

Expansion about the spherical case:

Plot

:

Evaluate numerically:

Out[1]=

Expansion about the spherical case:

Out[1]=

Plot

:

Out[1]=

Evaluate for complex arguments and parameters:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

For certain parameters SpheroidalPS evaluates exactly:

The different types of SpheroidalPS have different branch cut structures:

Solve the spheroidal differential equation in terms of SpheroidalPS:

Plot prolate and oblate versions of the same angular function:

SpheroidalPS is a band-limited function with bandwidth proportional to

:

Spheroidal angular harmonics are eigenfunctions of the Sinc transform on the interval

:

Spheroidal functions do not evaluate for half-integer values of

or generic values of

: