# SpheroidalPS

SpheroidalPS[n,m,γ,z]

gives the angular spheroidal function of the first kind.

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The angular spheroidal functions satisfy the differential equation with the spheroidal eigenvalue given by SpheroidalEigenvalue[n,m,γ].
• SpheroidalPS[n,m,0,z] is equivalent to LegendreP[n,m,z].
• SpheroidalPS[n,m,a,γ,z] gives spheroidal functions of type . The types are specified as for LegendreP.
• For certain special arguments, SpheroidalPS automatically evaluates to exact values.
• SpheroidalPS can be evaluated to arbitrary numerical precision.
• SpheroidalPS automatically threads over lists.

# Examples

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## Basic Examples(7)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

## Scope(19)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

### Specific Values(4)

Evaluate symbolically:

Find the first positive minimum of SpheroidalPS[4,0,1/2,x]:

Evaluate the SpheroidalPS function for half-integer parameters:

Different SpheroidalPS types give different symbolic forms:

### Visualization(3)

Plot the SpheroidalPS function for various orders:

Plot the real part of :

Plot the imaginary part of :

Types 2 and 3 of SpheroidalPS functions have different branch cut structures:

### Function Properties(4)

The real domain of SpheroidalPS:

The complex domain of SpheroidalPS:

SpheroidalPS is an even function with respect to γ:

SpheroidalPS has the mirror property :

### Differentiation(2)

The first derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when n=10, m=2 and γ=1/3:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The Taylor expansion at a generic point:

## Generalizations & Extensions(1)

The different types of SpheroidalPS have different branch cut structures:

## Applications(3)

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 :

For spheroidicity parameter :

For spheroidicity parameter , the bandwidth is higher:

## Properties & Relations(1)

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

## Possible Issues(2)

Spheroidal functions do not evaluate for half-integer values of or generic values of :

Angular spheroidal harmonics in the Wolfram Language adopt the MeixnerSchaefke normalization scheme:

Flammer normalization is also common:

Reconstruct table entries from Abramowitz and Stegun table 21.2: