# SpheroidalEigenvalue

SpheroidalEigenvalue[n,m,γ]

gives the spheroidal eigenvalue with degree and order .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• The spheroidal eigenvalues for successive correspond to the successive values of for which there exist normalizable solutions to the differential equation .
• SpheroidalEigenvalue[n,m,0]is equal to .
• For certain special arguments, SpheroidalEigenvalue automatically evaluates to exact values.
• SpheroidalEigenvalue can be evaluated to arbitrary numerical precision.
• SpheroidalEigenvalue automatically threads over lists.

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion in the spherical limit as γ approaches 0:

Series expansion at Infinity:

## Scope(13)

### 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(7)

Simple exact values are generated automatically:

Evaluate symbolically for integer parameters:

Evaluate symbolically for half-integer parameters:

Find the maximum of SpheroidalEigenvalue[1,2/3,x]:

SpheroidalEigenvalue evaluates exactly if m=1 and γ=n π/2:

### Visualization(2)

Plot the SpheroidalEigenvalue function for integer orders:

Plot the real part of :

Plot the imaginary part of :

## Applications(3)

Solve the spheroidal differential equation:

Solve this spheroidal-type differential equation:

Find a branch point of SpheroidalEigenvalue:

## Properties & Relations(1)

For half-integer values, the SpheroidalEigenvalue reduces to the MathieuCharacteristicA function:

## Possible Issues(1)

SpheroidalEigenvalue does not evaluate for half-integer or for generic :

The half-integer values of are singular for the near-spherical expansion:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_spheroidaleigenvalue, author="Wolfram Research", title="{SpheroidalEigenvalue}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalEigenvalue.html}", note=[Accessed: 13-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_spheroidaleigenvalue, organization={Wolfram Research}, title={SpheroidalEigenvalue}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalEigenvalue.html}, note=[Accessed: 13-July-2024 ]}