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MathieuCharacteristicExponent[a,q]

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

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Function Properties  
Applications  
Properties & Relations  
Neat Examples  
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Tech Notes
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MathieuCharacteristicExponent

MathieuCharacteristicExponent[a,q]

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

Details

Examples

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Basic Examples  (3)

Evaluate numerically:

Wolfram Language code: MathieuCharacteristicExponent[2, 0.5]

Plot over a subset of the reals:

Wolfram Language code: Plot[MathieuCharacteristicExponent[3, x], {x, -2, 2}]

Plot over a subset of the complexes:

Wolfram Language code: ComplexPlot3D[MathieuCharacteristicExponent[3, z], {z, -7 - 7I, 7 + 7I}, PlotLegends -> Automatic]

Scope  (15)

Numerical Evaluation  (7)

Evaluate numerically:

Wolfram Language code: MathieuCharacteristicExponent[0.2, 0.4]
Wolfram Language code: N[MathieuCharacteristicExponent[Pi, E]]

MathieuCharacteristicExponent threads elementwise over lists:

Wolfram Language code: MathieuCharacteristicExponent[5, {0.5, 1.0, 1.5}]

Evaluate to high precision:

Wolfram Language code: N[MathieuCharacteristicExponent[17 / 3, 1 / 2], 30]

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

Wolfram Language code: MathieuCharacteristicExponent[5, 0.501111111111004564560000]

Complex number inputs:

Wolfram Language code: N[MathieuCharacteristicExponent[Pi / 2, 2 + I]]

Evaluate efficiently at high precision:

Wolfram Language code: MathieuCharacteristicExponent[1 / 5647, 5`100]//Timing
Wolfram Language code: MathieuCharacteristicExponent[1 / 5271, 75`1000];//Timing

Compute average-case statistical intervals using Around:

Wolfram Language code: MathieuCharacteristicExponent[3, Around[1, 0.01]]

Compute the elementwise values of an array:

Wolfram Language code: MathieuCharacteristicExponent[.2, {{1, 0}, {0, 2}}]

Or compute the matrix MathieuCharacteristicExponent function using MatrixFunction:

Wolfram Language code: MatrixFunction[MathieuCharacteristicExponent[.2, #]&, {{1, 0}, {0, 2}}]

Specific Values  (2)

Simple exact values are generated automatically:

Wolfram Language code: MathieuCharacteristicExponent[a, 0]

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

Wolfram Language code: qval = q /. FindRoot[MathieuCharacteristicExponent[3, q ] == 1.7, {q, 1}]
Wolfram Language code: Plot[MathieuCharacteristicExponent[3, q ], {q, -1, 2}, Epilog -> Style[Point[{qval, MathieuCharacteristicExponent[3, qval ]}], PointSize[Large], Red]]

Visualization  (3)

Plot the MathieuCharacteristicExponent function for integer parameters:

Wolfram Language code: Plot[{MathieuCharacteristicExponent[3, x], MathieuCharacteristicExponent[5, x], MathieuCharacteristicExponent[7, x]}, {x, -2, 2}]

Plot the MathieuCharacteristicExponent function for noninteger parameters:

Wolfram Language code: Plot[{MathieuCharacteristicExponent[1 / 7, x], MathieuCharacteristicExponent[1 / 3, x], MathieuCharacteristicExponent[1 / 2, x]}, {x, -1, 1}]

Plot the real part of MathieuCharacteristicExponent:

Wolfram Language code: ComplexContourPlot[Re[MathieuCharacteristicExponent[10, z]], {z, -4 - 4I, 4 + 4I}, IconizedObject[«PlotOptions»]]

Plot the imaginary part of MathieuCharacteristicExponent:

Wolfram Language code: ComplexContourPlot[Im[MathieuCharacteristicExponent[10, z]], {z, -4 - 4I, 4 + 4I}, IconizedObject[«PlotOptions»]]

Function Properties  (3)

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

Wolfram Language code: FunctionMonotonicity[MathieuCharacteristicExponent[3, x], x]

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

Wolfram Language code: FunctionSign[MathieuCharacteristicExponent[3, x], x]

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

Wolfram Language code: FunctionConvexity[MathieuCharacteristicExponent[3, x], x]

Applications  (2)

Solve the Schrödinger equation with periodic potential:

Wolfram Language code: DSolve[-y''[x] + Cos[x]y[x] == ℰ y[x], y[x], x]

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:

Wolfram Language code: Plot[{ArcTan[Re[MathieuS[a, 2, 100]]], Im[MathieuCharacteristicExponent[a, 2]]}, {a, 2, 8}, WorkingPrecision -> 16, PlotStyle -> {{Dashing[Tiny]}, {Thick}}]

This shows the stability diagram for the Mathieu equation:

Wolfram Language code: RegionPlot[Im[MathieuCharacteristicExponent[a, q]] == 0, {q, 0, 20}, {a, -5, 20}, PlotPoints -> 60, MaxRecursion -> 3]

Properties & Relations  (2)

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

Wolfram Language code: FindRoot[MathieuCharacteristicExponent[a, 1] == 3 / 2, {a, 0.8, 1.6}]
Wolfram Language code: MathieuCharacteristicA[1.5, 1]

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

Wolfram Language code: Plot[{MathieuCharacteristicExponent[x, 0], Sqrt[x] + 1}, {x, 0, 5}]

Neat Examples  (1)

This shows the band gaps in a periodic potential:

Wolfram Language code: Plot3D[Im[MathieuCharacteristicExponent[a, q]], {a, -10, 15}, {q, 0, 6}]
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_2026_mathieucharacteristicexponent, author="Wolfram Research", title="{MathieuCharacteristicExponent}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuCharacteristicExponent.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_mathieucharacteristicexponent, organization={Wolfram Research}, title={MathieuCharacteristicExponent}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuCharacteristicExponent.html}, note=[Accessed: 13-June-2026]}