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MathieuCharacteristicB
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MathieuCharacteristicB

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

Details

Examples

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

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0gfpqt5nhsbp-nwc1dr
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0gfpqt5nhsbp-hq4a8h
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0gfpqt5nhsbp-kiedlx
Out[1]=1

Scope  (19)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[3]:=3
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https://wolfram.com/xid/0gfpqt5nhsbp-l274ju
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0gfpqt5nhsbp-cksbl4
Out[4]=4

Evaluate to high precision:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-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/0gfpqt5nhsbp-y7k4a
Out[2]=2

Complex number inputs:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-hfml09
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-di5gcr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0gfpqt5nhsbp-bq2c6r
Out[2]=2

Compute average-case statistical intervals using Around:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-cw18bq
Out[1]=1

Compute the elementwise values of an array:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-thgd2
Out[1]=1

Or compute the matrix MathieuCharacteristicB function using MatrixFunction:

In[2]:=2
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https://wolfram.com/xid/0gfpqt5nhsbp-o5jpo
Out[2]=2

Specific Values  (2)

Simple exact values are generated automatically:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-bmqd0y
Out[1]=1

Find the positive maximum of MathieuCharacteristicB[3,q]:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-f2hrld
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0gfpqt5nhsbp-bqf3y
Out[2]=2

Visualization  (3)

Plot the MathieuCharacteristicB function for integer parameters:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-ecj8m7
Out[1]=1

Plot the MathieuCharacteristicB function for noninteger parameters:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-b1j98m
Out[1]=1

Plot the real part of MathieuCharacteristicB:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-cjk9wl
Out[1]=1

Plot the imaginary part of MathieuCharacteristicB:

In[2]:=2
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https://wolfram.com/xid/0gfpqt5nhsbp-jujkdc
Out[2]=2

Function Properties  (6)

The real domain of MathieuCharacteristicB:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-jitxhe
Out[1]=1

TemplateBox[{r, x}, MathieuCharacteristicB] is a continuous function of :

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-m29jj1
Out[1]=1

TemplateBox[{1, x}, MathieuCharacteristicB] is neither non-increasing nor non-decreasing:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-xhpqv5
Out[1]=1

TemplateBox[{1, x}, MathieuCharacteristicB] is not injective:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-eljl3o
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0gfpqt5nhsbp-d29icx
Out[2]=2

MathieuCharacteristicB threads elementwise over lists:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-ysgfug
Out[1]=1

TraditionalForm formatting:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-cpcvv4

Series Expansions  (2)

Find the Taylor expansion using Series:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-ewr1h8
Out[1]=1

Plots of the first three approximations around :

In[2]:=2
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https://wolfram.com/xid/0gfpqt5nhsbp-binhar
Out[3]=3

Find the series expansion at infinity:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-jwxla7
Out[1]=1

Applications  (4)Sample problems that can be solved with this function

Symmetric periodic solutions of the Mathieu differential equation:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-gbpcnu
Out[1]=1

This shows the stability diagram for the Mathieu equation:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-bwxcw2
Out[1]=1

As a function of the first argument, MathieuCharacteristicB is a piecewise continuous function (called bands and band gaps in solid-state physics):

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-f1cz69
Out[1]=1

Solve the Laplace equation in an ellipse using separation of variables:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-v8ojo

This finds a zero:

In[2]:=2
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https://wolfram.com/xid/0gfpqt5nhsbp-dg5z3
Out[2]=2

This plots an eigenfunction. It vanishes at the ellipse boundary:

In[3]:=3
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https://wolfram.com/xid/0gfpqt5nhsbp-jz6sc
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

There is no zero-order MathieuCharacteristicB:

In[1]:=1
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https://wolfram.com/xid/0gfpqt5nhsbp-dwy1z0
Out[1]=1

Neat Examples  (1)Surprising or curious use cases

Branch points of the Mathieu characteristic along the imaginary q axis:

In[1]:=1
✖
https://wolfram.com/xid/0gfpqt5nhsbp-bq6uj
Out[1]=1
Wolfram Research (1996), MathieuCharacteristicB, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html.
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Wolfram Research (1996), MathieuCharacteristicB, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html.

Text

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

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

CMS

Wolfram Language. 1996. "MathieuCharacteristicB." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html.

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Wolfram Language. 1996. "MathieuCharacteristicB." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html.

APA

Wolfram Language. (1996). MathieuCharacteristicB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html

✖
Wolfram Language. (1996). MathieuCharacteristicB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html

BibTeX

@misc{reference.wolfram_2025_mathieucharacteristicb, author="Wolfram Research", title="{MathieuCharacteristicB}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html}", note=[Accessed: 17-May-2025 ]}

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@misc{reference.wolfram_2025_mathieucharacteristicb, author="Wolfram Research", title="{MathieuCharacteristicB}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html}", note=[Accessed: 17-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_mathieucharacteristicb, organization={Wolfram Research}, title={MathieuCharacteristicB}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html}, note=[Accessed: 17-May-2025 ]}

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@online{reference.wolfram_2025_mathieucharacteristicb, organization={Wolfram Research}, title={MathieuCharacteristicB}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html}, note=[Accessed: 17-May-2025 ]}