MathieuCharacteristicExponent
✖
MathieuCharacteristicExponent
gives the characteristic exponent r for Mathieu functions with characteristic value a and parameter q.
Details
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- 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 allclose allBasic Examples (3)Summary of the most common use cases
Scope (15)Survey of the scope of standard use cases
Numerical Evaluation (7)
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https://wolfram.com/xid/0tp309i643p23m-l274ju
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https://wolfram.com/xid/0tp309i643p23m-whe1w
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MathieuCharacteristicExponent threads elementwise over lists:
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https://wolfram.com/xid/0tp309i643p23m-dkfve
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https://wolfram.com/xid/0tp309i643p23m-b0wt9
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The precision of the output tracks the precision of the input:
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https://wolfram.com/xid/0tp309i643p23m-y7k4a
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https://wolfram.com/xid/0tp309i643p23m-hfml09
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Evaluate efficiently at high precision:
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https://wolfram.com/xid/0tp309i643p23m-di5gcr
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https://wolfram.com/xid/0tp309i643p23m-bq2c6r
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Compute average-case statistical intervals using Around:
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https://wolfram.com/xid/0tp309i643p23m-cw18bq
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Compute the elementwise values of an array:
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https://wolfram.com/xid/0tp309i643p23m-thgd2
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Or compute the matrix MathieuCharacteristicExponent function using MatrixFunction:
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https://wolfram.com/xid/0tp309i643p23m-o5jpo
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Specific Values (2)
Simple exact values are generated automatically:
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https://wolfram.com/xid/0tp309i643p23m-bmqd0y
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Find a value of q for which MathieuCharacteristicExponent[3,q]=1.7:
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https://wolfram.com/xid/0tp309i643p23m-f2hrld
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https://wolfram.com/xid/0tp309i643p23m-fo3om1
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Visualization (3)
Plot the MathieuCharacteristicExponent function for integer parameters:
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https://wolfram.com/xid/0tp309i643p23m-ecj8m7
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Plot the MathieuCharacteristicExponent function for noninteger parameters:
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https://wolfram.com/xid/0tp309i643p23m-b1j98m
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Plot the real part of MathieuCharacteristicExponent:
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https://wolfram.com/xid/0tp309i643p23m-cjk9wl
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Plot the imaginary part of MathieuCharacteristicExponent:
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https://wolfram.com/xid/0tp309i643p23m-earwdu
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Function Properties (3)
MathieuCharacteristicExponent[3,x] is neither non-decreasing nor non-increasing:
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https://wolfram.com/xid/0tp309i643p23m-2ra8g
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MathieuCharacteristicExponent[3,x] is neither non-negative nor non-positive:
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https://wolfram.com/xid/0tp309i643p23m-dvzykj
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MathieuCharacteristicExponent[3,x] is neither convex nor concave:
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https://wolfram.com/xid/0tp309i643p23m-l0srvu
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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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https://wolfram.com/xid/0tp309i643p23m-c2x80s
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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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https://wolfram.com/xid/0tp309i643p23m-c1gowv
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This shows the stability diagram for the Mathieu equation:
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https://wolfram.com/xid/0tp309i643p23m-ba9xjy
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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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https://wolfram.com/xid/0tp309i643p23m-pws0wl
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https://wolfram.com/xid/0tp309i643p23m-dmcwqu
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From the plot, you can see that MathieuCharacteristicExponent[x,0]=:
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https://wolfram.com/xid/0tp309i643p23m-dn1iet
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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
]}
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
]}