# SpheroidalQS

SpheroidalQS[n,m,γ,z]

gives the angular spheroidal function of the second kind.

# Details • 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].
• SpheroidalQS[n,m,0,z] is equivalent to LegendreQ[n,m,z].
• SpheroidalQS[n,m,a,γ,z] gives spheroidal functions of type . The types are specified as for LegendreP.
• For certain special arguments, SpheroidalQS automatically evaluates to exact values.
• SpheroidalQS can be evaluated to arbitrary numerical precision.
• SpheroidalQS automatically threads over lists.

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion at the origin:

## Scope(21)

### Numerical Evaluation(5)

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)

SpheroidalQS[n,m,0,x] is equivalent to the LegendreQ[n,m,x] function:

Find the first positive maximum of SpheroidalQS[4,0,1/2,x]:

The SpheroidalQS function is equal to zero for half-integer parameters:

Different SpheroidalQS types give different symbolic forms:

### Visualization(3)

Plot the SpheroidalQS function for various orders:

Plot the real part of :

Plot the imaginary part of :

Types 2 and 3 of SpheroidalQS functions have different branch cut structures:

### Function Properties(5) is an even function: has both singularities and discontinuities for : is neither non-decreasing nor non-increasing: is neither non-negative nor non-positive:

### Differentiation(2)

The first derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when n=5, m=2 and γ=1:

### 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 SpheroidalQS have different branch cut structures:

## Applications(2)

Solve the spheroidal differential equation in terms of SpheroidalQS:

Plot prolate and oblate versions of the same angular function:

## Possible Issues(1)

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