# 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(4)

Evaluate numerically:

Expansion about the spherical case:

Plot over a subset of the reals:

Series expansion at the origin:

## Scope(20)

### 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:

TraditionalForm formatting:

### Visualization(2)

Plot the SpheroidalQSPrime function for various orders:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(2)

has both singularities and discontinuities for :

SpheroidalQSPrime is neither non-negative nor non-positive:

### 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 :

Wolfram Research (2007), SpheroidalQSPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalQSPrime.html.

#### Text

Wolfram Research (2007), SpheroidalQSPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalQSPrime.html.

#### CMS

Wolfram Language. 2007. "SpheroidalQSPrime." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SpheroidalQSPrime.html.

#### APA

Wolfram Language. (2007). SpheroidalQSPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SpheroidalQSPrime.html

#### BibTeX

@misc{reference.wolfram_2024_spheroidalqsprime, author="Wolfram Research", title="{SpheroidalQSPrime}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalQSPrime.html}", note=[Accessed: 15-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_spheroidalqsprime, organization={Wolfram Research}, title={SpheroidalQSPrime}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalQSPrime.html}, note=[Accessed: 15-July-2024 ]}