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

open allclose all

Basic Examples  (7)

Evaluate numerically:

Expansion about the spherical case:

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 TemplateBox[{3, 0, 1, {x, +, {i,  , y}}}, SpheroidalPS]:

Plot the imaginary part of TemplateBox[{3, 0, 1, {x, +, {i,  , y}}}, SpheroidalPS]:

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 TemplateBox[{1, 2, 3, {z, }}, SpheroidalPS]=TemplateBox[{1, 2, 3, z}, SpheroidalPS]:

TraditionalForm formatting:

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 n or generic values of m:

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:

Introduced in 2007
 (6.0)