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Built-in Mathematica Symbol

SpheroidalQS

, z]
gives the angular spheroidal function

of the second kind.

The angular spheroidal functions satisfy the differential equation with the spheroidal eigenvalue given by SpheroidalEigenvalue[n, m, ].

SpheroidalQS[n, m, a, , z] gives spheroidal functions of type . The types are specified as for LegendreP.

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

Evaluate numerically:

Expansion about the spherical case:

Plot

Evaluate numerically:

Expansion about the spherical case:

Plot

Evaluate for complex arguments and parameters:

Evaluate to high precision:

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

For certain parameters SpheroidalQS evaluates exactly:

The different types of SpheroidalQS have different branch cut structures:

Solve the spheroidal differential equation in terms of SpheroidalQS:

Plot prolate and oblate versions of the same angular function:

Spheroidal functions do not generically evaluate for half-integer values of