# SpheroidalQSPrime

SpheroidalQSPrime[n,m,γ,z]

gives the derivative with respect to of the angular spheroidal function of the second kind.

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• SpheroidalQSPrime[n,m,a,γ,z] uses spheroidal functions of type . The types are specified as for SpheroidalPS.
• For certain special arguments, SpheroidalQSPrime automatically evaluates to exact values.
• SpheroidalQSPrime can be evaluated to arbitrary numerical precision.
• SpheroidalQSPrime automatically threads over lists.

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

## Scope(18)

### Numerical Evaluation(4)

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(5)

Evaluate symbolically:

Find the first positive minimum of SpheroidalQSPrime[4,0,1/2,x]:

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

Different SpheroidalQSPrime types give different symbolic forms:

### Visualization(2)

Plot the SpheroidalQSPrime function for various orders:

Plot the real part of :

Plot the imaginary part of :

### Differentiation(2)

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:

### Integration(3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

## Applications(1)

Plot prolate and oblate versions of the same angular function:

## Possible Issues(1)

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

Introduced in 2007
(6.0)