MathieuCharacteristicExponent [a,q]
gives the characteristic exponent r for Mathieu functions with characteristic value a and parameter q.
MathieuCharacteristicExponent
MathieuCharacteristicExponent [a,q]
gives the characteristic exponent r for Mathieu functions with characteristic value a and parameter q.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- All Mathieu functions have the form where has period and r is the Mathieu characteristic exponent.
- For certain special arguments, MathieuCharacteristicExponent automatically evaluates to exact values.
- MathieuCharacteristicExponent can be evaluated to arbitrary numerical precision.
- MathieuCharacteristicExponent automatically threads over lists.
Examples
open all close allBasic 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:
Tech Notes
Related Guides
Related Links
History
Introduced in 1996 (3.0)
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: 17-November-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: 17-November-2025]}