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MathieuCharacteristicB[r,q]

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

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Function Properties  
Series Expansions  
Applications  
Possible Issues  
Neat Examples  
See Also
Tech Notes
Related Guides
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MathieuCharacteristicB

MathieuCharacteristicB[r,q]

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

Details

Examples

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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 MathieuCharacteristicB[3,q]:

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:

TemplateBox[{r, x}, MathieuCharacteristicB] is a continuous function of :

TemplateBox[{1, x}, MathieuCharacteristicB] is neither non-increasing nor non-decreasing:

TemplateBox[{1, x}, MathieuCharacteristicB] is not injective:

MathieuCharacteristicB threads elementwise over lists:

TraditionalForm formatting:

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_2025_mathieucharacteristicb, author="Wolfram Research", title="{MathieuCharacteristicB}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html}", note=[Accessed: 07-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_mathieucharacteristicb, organization={Wolfram Research}, title={MathieuCharacteristicB}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html}, note=[Accessed: 07-August-2025]}