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BUILT-IN MATHEMATICA SYMBOL

SpheroidalEigenvalue

gives the spheroidal eigenvalue with degree

and order

The spheroidal eigenvalues for successive correspond to the successive values of for which there exist normalizable solutions to the differential equation .

For certain special arguments, SpheroidalEigenvalue automatically evaluates to exact values.

Evaluate numerically:

Series expansion in the spherical limit:

Evaluate numerically:

Out[1]=

Series expansion in the spherical limit:

Out[1]=

Evaluate for complex arguments:

Evaluate to high precision:

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

SpheroidalEigenvalue threads element-wise over lists:

For certain parameters SpheroidalEigenvalue evaluates exactly:

Solve the spheroidal differential equation:

Find a branch point of SpheroidalEigenvalue:

SpheroidalEigenvalue does not evaluate for half-integer

or for generic

The half-integer values of

are singular for the near-spherical expansion: