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MathieuCharacteristicExponent
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MathieuCharacteristicExponent

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

Details

Examples

open allclose all

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

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0tp309i643p23m-ejytpd
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-j2hu8f
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-h680z
Out[1]=1

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

Numerical Evaluation  (7)

Evaluate numerically:

In[2]:=2
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https://wolfram.com/xid/0tp309i643p23m-l274ju
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0tp309i643p23m-whe1w
Out[3]=3

MathieuCharacteristicExponent threads elementwise over lists:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-dkfve
Out[1]=1

Evaluate to high precision:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-b0wt9
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0tp309i643p23m-y7k4a
Out[2]=2

Complex number inputs:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-hfml09
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-di5gcr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tp309i643p23m-bq2c6r
Out[2]=2

Compute average-case statistical intervals using Around:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-cw18bq
Out[1]=1

Compute the elementwise values of an array:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-thgd2
Out[1]=1

Or compute the matrix MathieuCharacteristicExponent function using MatrixFunction:

In[2]:=2
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https://wolfram.com/xid/0tp309i643p23m-o5jpo
Out[2]=2

Specific Values  (2)

Simple exact values are generated automatically:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-bmqd0y
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-f2hrld
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tp309i643p23m-fo3om1
Out[2]=2

Visualization  (3)

Plot the MathieuCharacteristicExponent function for integer parameters:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-ecj8m7
Out[1]=1

Plot the MathieuCharacteristicExponent function for noninteger parameters:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-b1j98m
Out[1]=1

Plot the real part of MathieuCharacteristicExponent:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-cjk9wl
Out[1]=1

Plot the imaginary part of MathieuCharacteristicExponent:

In[2]:=2
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https://wolfram.com/xid/0tp309i643p23m-earwdu
Out[2]=2

Function Properties  (3)

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

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-2ra8g
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-dvzykj
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-l0srvu
Out[1]=1

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

Solve the Schrödinger equation with periodic potential:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-c2x80s
Out[1]=1

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:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-c1gowv
Out[1]=1

This shows the stability diagram for the Mathieu equation:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-ba9xjy
Out[1]=1

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

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

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-pws0wl
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tp309i643p23m-dmcwqu
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-dn1iet
Out[1]=1

Neat Examples  (1)Surprising or curious use cases

This shows the band gaps in a periodic potential:

In[1]:=1
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https://wolfram.com/xid/0tp309i643p23m-dr729i
Out[1]=1
Wolfram Research (1996), MathieuCharacteristicExponent, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuCharacteristicExponent.html.
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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.

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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.

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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

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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: 29-March-2025 ]}

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@misc{reference.wolfram_2025_mathieucharacteristicexponent, author="Wolfram Research", title="{MathieuCharacteristicExponent}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuCharacteristicExponent.html}", note=[Accessed: 29-March-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: 29-March-2025 ]}

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@online{reference.wolfram_2025_mathieucharacteristicexponent, organization={Wolfram Research}, title={MathieuCharacteristicExponent}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuCharacteristicExponent.html}, note=[Accessed: 29-March-2025 ]}