# SpheroidalS1

SpheroidalS1[n,m,γ,z]

gives the radial spheroidal function of the first kind.

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• The radial spheroidal functions satisfy the differential equation with the spheroidal eigenvalue given by SpheroidalEigenvalue[n,m,γ].
• The are normalized according to the MeixnerSchäfke scheme.
• SpheroidalS1 can be evaluated to arbitrary numerical precision.
• SpheroidalS1 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:

Series expansion at a singular point:

## 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(4)

Simple exact values are generated automatically:

Find the first positive maximum of SpheroidalS1[2,0,5,x]:

SpheroidalS1 functions become elementary if m=1 and γ=n π/2 :

### Visualization(3)

Plot the SpheroidalS1 function for integer orders:

Plot the SpheroidalS1 function for noninteger parameters:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(5)

SpheroidalS1 is not an analytic function:

has both singularities and discontinuities for :

is neither non-decreasing nor non-increasing:

is not injective:

SpheroidalS1 is neither non-negative nor non-positive:

SpheroidalS1 is neither convex nor concave:

### Differentiation(2)

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:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

## Applications(4)

Spheroidal angular harmonics are eigenfunctions of the Sinc transform on the interval :

Plot the eigenvalue:

Find resonant frequencies for the Dirichlet problem in the prolate spheroidal cavity:

Determine the first few frequencies:

Plot the prolate and oblate functions:

Build a near-spherical approximation to :

First few terms of the approximation:

Compare numerically:

## Possible Issues(1)

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

Wolfram Research (2007), SpheroidalS1, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalS1.html.

#### Text

Wolfram Research (2007), SpheroidalS1, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalS1.html.

#### CMS

Wolfram Language. 2007. "SpheroidalS1." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SpheroidalS1.html.

#### APA

Wolfram Language. (2007). SpheroidalS1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SpheroidalS1.html

#### BibTeX

@misc{reference.wolfram_2024_spheroidals1, author="Wolfram Research", title="{SpheroidalS1}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalS1.html}", note=[Accessed: 20-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_spheroidals1, organization={Wolfram Research}, title={SpheroidalS1}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalS1.html}, note=[Accessed: 20-July-2024 ]}