# MathieuCharacteristicB

gives the characteristic value for odd Mathieu functions with characteristic exponent r and parameter q.

# Examples

open allclose all

## Basic Examples(3)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

## Scope(19)

### Numerical Evaluation(6)

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:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix MathieuCharacteristicB function using MatrixFunction:

### Specific Values(2)

Simple exact values are generated automatically:

Find the positive maximum of :

### Visualization(3)

Plot the MathieuCharacteristicB function for integer parameters:

Plot the MathieuCharacteristicB function for noninteger parameters:

Plot the real part of MathieuCharacteristicB:

Plot the imaginary part of MathieuCharacteristicB:

### Function Properties(6)

The real domain of MathieuCharacteristicB:

is a continuous function of :

is neither non-increasing nor non-decreasing:

is not injective:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at infinity:

## Applications(4)

Symmetric periodic solutions of the Mathieu differential equation:

This shows the stability diagram for the Mathieu equation:

As a function of the first argument, MathieuCharacteristicB is a piecewise continuous function (called bands and band gaps in solid-state physics):

Solve the Laplace equation in an ellipse using separation of variables:

This finds a zero:

This plots an eigenfunction. It vanishes at the ellipse boundary:

## Possible Issues(1)

There is no zero-order MathieuCharacteristicB:

## Neat Examples(1)

Branch points of the Mathieu characteristic along the imaginary q axis:

Wolfram Research (1996), MathieuCharacteristicB, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html.

#### Text

Wolfram Research (1996), MathieuCharacteristicB, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html.

#### CMS

Wolfram Language. 1996. "MathieuCharacteristicB." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html.

#### APA

Wolfram Language. (1996). MathieuCharacteristicB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html

#### BibTeX

@misc{reference.wolfram_2024_mathieucharacteristicb, author="Wolfram Research", title="{MathieuCharacteristicB}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_mathieucharacteristicb, organization={Wolfram Research}, title={MathieuCharacteristicB}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html}, note=[Accessed: 14-September-2024 ]}