gives the characteristic value 
for odd Mathieu functions with characteristic exponent r and parameter q.
MathieuCharacteristicB
gives the characteristic value 
for odd Mathieu functions with characteristic exponent r and parameter q.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The characteristic value
gives the value of the parameter 
in 
for which the solution has the form 
where 
is an odd function of 
with period 
. - When r is not a real integer, MathieuCharacteristicB gives the same results as MathieuCharacteristicA.
- For certain special arguments, MathieuCharacteristicB automatically evaluates to exact values.
- MathieuCharacteristicB can be evaluated to arbitrary numerical precision.
- MathieuCharacteristicB automatically threads over lists.
Examples
open all close allBasic Examples (3)
MathieuCharacteristicB[1, 0.5]Plot over a subset of the reals:
Plot[MathieuCharacteristicB[3, x], {x, -4, 4}]Plot over a subset of the complexes:
ComplexPlot3D[MathieuCharacteristicB[1 / 3, z], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]Scope (19)
Numerical Evaluation (6)
N[MathieuCharacteristicB[Pi / 2, Pi / 4]]MathieuCharacteristicB[Pi, 0.5]N[MathieuCharacteristicB[15 / 17, 5 / 3], 25]The precision of the output tracks the precision of the input:
MathieuCharacteristicB[4.21111111111110000111, 5]N[MathieuCharacteristicB[Pi / 5, 5 + I]]Evaluate efficiently at high precision:
MathieuCharacteristicB[1 / 347, 5`100]//TimingMathieuCharacteristicB[152 / 71, 5`1000];//TimingCompute average-case statistical intervals using Around:
MathieuCharacteristicB[1, Around[1 / 2, 0.01]]Compute the elementwise values of an array:
MathieuCharacteristicB[.2, {{1 / 2, 0}, {0, 1 / 2}}]Or compute the matrix MathieuCharacteristicB function using MatrixFunction:
MatrixFunction[MathieuCharacteristicB[.2, #]&, {{1 / 2, 0}, {0, 1 / 2}}]Specific Values (2)
Simple exact values are generated automatically:
MathieuCharacteristicB[r, 0]Find the positive maximum of MathieuCharacteristicB[3,q]:
qmax = q /. FindRoot[D[MathieuCharacteristicB[3, q ], q] == 0, {q, 4}]Plot[MathieuCharacteristicB[3, q ], {q, -1, 30}, Epilog -> Style[Point[{qmax, MathieuCharacteristicB[3, qmax ]}], PointSize[Large], Red]]Visualization (3)
Plot the MathieuCharacteristicB function for integer parameters:
Plot[{MathieuCharacteristicB[3, x], MathieuCharacteristicB[2, x], MathieuCharacteristicB[1, x]}, {x, -10, 10}]Plot the MathieuCharacteristicB function for noninteger parameters:
Plot[{MathieuCharacteristicB[1 / 7, x], MathieuCharacteristicB[1 / 3, x], MathieuCharacteristicB[1 / 2, x]}, {x, -1, 1}]Plot the real part of MathieuCharacteristicB:
ComplexContourPlot[Re[MathieuCharacteristicB[10, z]], {z, -4 - 4I, 4 + 4I}, IconizedObject[«PlotOptions»]]Plot the imaginary part of MathieuCharacteristicB:
ComplexContourPlot[Im[MathieuCharacteristicB[10, z]], {z, -4 - 4I, 4 + 4I}, IconizedObject[«PlotOptions»]]Function Properties (6)
The real domain of MathieuCharacteristicB:
FunctionDomain[MathieuCharacteristicB[r, x], x]![TemplateBox[{r, x}, MathieuCharacteristicB]](Files/MathieuCharacteristicB.en/9.png "TemplateBox[{r, x}, MathieuCharacteristicB]"){kind=small}
FunctionContinuous[MathieuCharacteristicB[r, x], x, Assumptions -> r ≠ 0 && r∈Reals]![TemplateBox[{1, x}, MathieuCharacteristicB]](Files/MathieuCharacteristicB.en/11.png "TemplateBox[{1, x}, MathieuCharacteristicB]"){kind=small}
is neither non-increasing nor non-decreasing:
FunctionMonotonicity[MathieuCharacteristicB[1, x], x]![TemplateBox[{1, x}, MathieuCharacteristicB]](Files/MathieuCharacteristicB.en/12.png "TemplateBox[{1, x}, MathieuCharacteristicB]"){kind=small}
FunctionInjective[MathieuCharacteristicB[1, x], x]Plot[{MathieuCharacteristicB[1, x], 1}, {x, -7, 3}]MathieuCharacteristicB threads elementwise over lists:
MathieuCharacteristicB[{1, 2, 3, 4, 5}, 0.33]TraditionalForm formatting:
TraditionalForm[MathieuCharacteristicB[r, x]]Series Expansions (2)
Find the Taylor expansion using Series:
Series[MathieuCharacteristicB[2, q], {q, 0, 3}]Plots of the first three approximations around 
:
terms = Normal@Table[Series[MathieuCharacteristicB[2, q], {q, 0, m}], {m, 1, 5, 2}];
Plot[{MathieuCharacteristicB[2, q], terms}//Evaluate, {q, -9, 9}]Find the series expansion at infinity:
Series[MathieuCharacteristicB[4 / 3, q], {q, ∞, 3}]Applications (4)
Symmetric periodic solutions of the Mathieu differential equation:
Plot[Evaluate@Table[MathieuS[MathieuCharacteristicB[r, 2], 2, x], {r, 3}], {x, -6, 6}]This shows the stability diagram for the Mathieu equation:
Plot[Evaluate[Flatten@Table[{MathieuCharacteristicA[r, q],
MathieuCharacteristicB[r + 1, q]}, {r, 0, 4}]], {q, 0, 20}, Filling -> Table[i -> {i + 1}, {i, 1, 10, 2}]]As a function of the first argument, MathieuCharacteristicB is a piecewise continuous function (called bands and band gaps in solid-state physics):
Plot[MathieuCharacteristicB[r, 1], {r, -2.5, 2.5}]Solve the Laplace equation in an ellipse using separation of variables:
ellipseψ[n_, z0_][r_, f_] := With[{ch = MathieuCharacteristicB[n, z0]}, I MathieuS[ch, z0, I r]MathieuS[ch, z0, f]]FindRoot[MathieuS[MathieuCharacteristicB[5, q], q, I / 2], {q, 30}, WorkingPrecision -> 25]This plots an eigenfunction. It vanishes at the ellipse boundary:
ParametricPlot3D[{Cosh[r] Cos[f], Sinh[r] Sin[f], 2ellipseψ[5, q /. %][r, f]}, {r, 0, 1 / 2}, {f, 0, 2Pi}, Mesh -> False]Possible Issues (1)
There is no zero-order MathieuCharacteristicB:
MathieuCharacteristicB[0, 0.5]
Related Links
MathWorld
The Wolfram Functions SiteText
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.
APA
Wolfram Language. (1996). MathieuCharacteristicB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MathieuCharacteristicB.html