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 Meixner–Schäfke scheme.
- SpheroidalS1 can be evaluated to arbitrary numerical precision.
- SpheroidalS1 automatically threads over lists.
Examplesopen allclose all
Basic Examples (5)
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:
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)
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 :
Plot the SpheroidalS1 function for integer orders:
Plot the SpheroidalS1 function for noninteger parameters:
Plot the real part of :
Plot the imaginary part of :
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:
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 and generic values of :