gives the angular spheroidal function of the first kind.
- 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.
Examplesopen allclose all
Basic Examples (7)
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:
Numerical Evaluation (4)
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)
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:
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:
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:
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 Meixner–Schaefke normalization scheme:
Flammer normalization is also common:
Reconstruct table entries from Abramowitz and Stegun table 21.2: