gives the radial spheroidal function of the first kind.


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The radial spheroidal functions satisfy the differential equation with the spheroidal eigenvalue given by SpheroidalEigenvalue[n,m,γ].
  • The are normalized according to the MeixnerSchäfke scheme.
  • SpheroidalS1 can be evaluated to arbitrary numerical precision.
  • SpheroidalS1 automatically threads over lists.


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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at a singular point:

Scope  (15)

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)

Simple exact values are generated automatically:

Find the first positive maximum of SpheroidalS1[2,0,5,x]:

SpheroidalS1 functions become elementary if m=1 and γ=n π/2 :

TraditionalForm typesetting:

Visualization  (3)

Plot the SpheroidalS1 function for integer orders:

Plot the SpheroidalS1 function for noninteger parameters:

Plot the real part of TemplateBox[{2, 0, 1, {x, +, {i,  , y}}}, SpheroidalS1]:

Plot the imaginary part of TemplateBox[{2, 0, 1, {x, +, {i,  , y}}}, SpheroidalS1]:

Differentiation  (2)

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 :

Taylor expansion at a generic point:

Applications  (3)

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

Plot the eigenvalue:

Find resonant frequencies for the Dirichlet problem in the prolate spheroidal cavity:

Determine the first few frequencies:

Plot the prolate and oblate functions:

Possible Issues  (1)

Spheroidal functions do not evaluate for half-integer values of n and generic values of m:

Introduced in 2007