# SpheroidalPSPrime

SpheroidalPSPrime[n,m,γ,z]

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

# Details

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

# Examples

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## Basic Examples(6)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

## Scope(26)

### 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(4)

Evaluate symbolically:

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

Evaluate the SpheroidalPSPrime function for half-integer parameters:

Different SpheroidalPSPrime types give different symbolic forms:

### Visualization(3)

Plot the SpheroidalPSPrime function for various orders:

Plot the real part of :

Plot the imaginary part of :

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

### Function Properties(8)

The real domain of :

The complex domain of :

is an even function with respect to :

has the mirror property :

has no singularities or discontinuities:

is neither non-decreasing nor non-increasing:

is not injective:

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=10, m=2 and γ=1/3:

### 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 :

The Taylor expansion at a generic point:

## Generalizations & Extensions(1)

SpheroidalPSPrime of different types have different branch cut structures:

## Applications(1)

Plot prolate and oblate versions of the same angular function:

## Possible Issues(1)

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

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2022_spheroidalpsprime, author="Wolfram Research", title="{SpheroidalPSPrime}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalPSPrime.html}", note=[Accessed: 29-May-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_spheroidalpsprime, organization={Wolfram Research}, title={SpheroidalPSPrime}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalPSPrime.html}, note=[Accessed: 29-May-2023 ]}