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

Details

Examples

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Basic Examples  (3)Summary of the most common use cases

Evaluate numerically:

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Plot over a subset of the reals:

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Plot over a subset of the complexes:

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Scope  (15)Survey of the scope of standard use cases

Numerical Evaluation  (7)

Evaluate numerically:

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MathieuCharacteristicExponent threads elementwise over lists:

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Evaluate to high precision:

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The precision of the output tracks the precision of the input:

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Complex number inputs:

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Evaluate efficiently at high precision:

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Out[2]=2

Compute average-case statistical intervals using Around:

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Compute the elementwise values of an array:

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Or compute the matrix MathieuCharacteristicExponent function using MatrixFunction:

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Specific Values  (2)

Simple exact values are generated automatically:

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

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Out[2]=2

Visualization  (3)

Plot the MathieuCharacteristicExponent function for integer parameters:

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Plot the MathieuCharacteristicExponent function for noninteger parameters:

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Plot the real part of MathieuCharacteristicExponent:

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Plot the imaginary part of MathieuCharacteristicExponent:

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Function Properties  (3)

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

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

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MathieuCharacteristicExponent[3,x] is neither convex nor concave:

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Applications  (2)Sample problems that can be solved with this function

Solve the Schrödinger equation with periodic potential:

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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:

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This shows the stability diagram for the Mathieu equation:

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Properties & Relations  (2)Properties of the function, and connections to other functions

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

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From the plot, you can see that MathieuCharacteristicExponent[x,0]=:

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Neat Examples  (1)Surprising or curious use cases

This shows the band gaps in a periodic potential:

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

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.

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

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: 24-February-2025 ]}

@misc{reference.wolfram_2025_mathieucharacteristicexponent, author="Wolfram Research", title="{MathieuCharacteristicExponent}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuCharacteristicExponent.html}", note=[Accessed: 24-February-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: 24-February-2025 ]}

@online{reference.wolfram_2025_mathieucharacteristicexponent, organization={Wolfram Research}, title={MathieuCharacteristicExponent}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuCharacteristicExponent.html}, note=[Accessed: 24-February-2025 ]}