MathieuCharacteristicExponent

MathieuCharacteristicExponent[a,q]

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

Details

Examples

open allclose all

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:

MathieuCharacteristicExponent threads elementwise over lists:

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)

MathieuCharacteristicExponent[3,x] is neither non-decreasing nor non-increasing:

MathieuCharacteristicExponent[3,x] is neither non-negative nor non-positive:

MathieuCharacteristicExponent[3,x] 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 MathieuCharacteristicExponent[x,0]=:

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_2025_mathieucharacteristicexponent, author="Wolfram Research", title="{MathieuCharacteristicExponent}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuCharacteristicExponent.html}", note=[Accessed: 23-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_mathieucharacteristicexponent, organization={Wolfram Research}, title={MathieuCharacteristicExponent}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuCharacteristicExponent.html}, note=[Accessed: 23-April-2025 ]}