# MathieuCharacteristicExponent

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

# 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(15)

### Numerical Evaluation(7)

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 MathieuCharacteristicExponent function using MatrixFunction:

### Specific Values(2)

Simple exact values are generated automatically:

Find a value of q for which MathieuCharacteristicExponent[3,q]=1.7:

### Visualization(3)

Plot the MathieuCharacteristicExponent function for integer parameters:

Plot the MathieuCharacteristicExponent function for noninteger parameters:

Plot the real part of MathieuCharacteristicExponent:

Plot the imaginary part of MathieuCharacteristicExponent:

### Function Properties(3)

is neither non-decreasing nor non-increasing:

is neither non-negative nor non-positive:

is neither convex nor concave:

## Applications(2)

Solve the Schrödinger equation with periodic potential:

By the Bloch theorem, solutions are bounded provided is within an energy band. The energy gap corresponds to a range of where MathieuCharacteristicExponent has a non-vanishing imaginary part:

This shows the stability diagram for the Mathieu equation:

## Properties & Relations(2)

The characteristic exponent and the characteristic are inverses of each other:

From the plot, you can see that :

## Neat Examples(1)

This shows the band gaps in a periodic potential:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_mathieucharacteristicexponent, author="Wolfram Research", title="{MathieuCharacteristicExponent}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuCharacteristicExponent.html}", note=[Accessed: 12-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_mathieucharacteristicexponent, organization={Wolfram Research}, title={MathieuCharacteristicExponent}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuCharacteristicExponent.html}, note=[Accessed: 12-September-2024 ]}