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LameEigenvalueB[ν,j,m]

gives the ^(th) Lamé eigenvalue TemplateBox[{nu, j, m}, LameEigenvalueB] of order with elliptic parameter for the Lamé function LameS[ν,j,z,m].

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Series Expansions  
Function Representations  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
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LameEigenvalueB

LameEigenvalueB[ν,j,m]

gives the ^(th) Lamé eigenvalue TemplateBox[{nu, j, m}, LameEigenvalueB] of order with elliptic parameter for the Lamé function LameS[ν,j,z,m].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Lamé eigenvalue TemplateBox[{nu, j, m}, LameEigenvalueB] for successive gives the value of the parameter in the Lamé differential equation (where TemplateBox[{z, m}, JacobiSN] is the Jacobi elliptic function JacobiSN[z,m]), for which the solution is the function LameS[ν,j,z,m].
  • For certain special arguments, LameEigenvalueB automatically evaluates to exact values.
  • LameEigenvalueB[ν,j,0]=j2.
  • LameEigenvalueB can be evaluated to arbitrary numerical precision.
  • LameEigenvalueB automatically threads over lists.

Examples

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Basic Examples  (2)

Evaluate numerically:

Wolfram Language code: LameEigenvalueB[0.9, 2, 0.3]

Plot the LameEigenvalueB function:

Wolfram Language code: Plot[LameEigenvalueB[ν, 2, 0.5], {ν, -1 / 2, 6}]

Scope  (14)

Numerical Evaluation  (5)

Evaluate to high precision:

Wolfram Language code: N[LameEigenvalueB[9 / 10, 2, 3 / 10], 50]

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

Wolfram Language code: LameEigenvalueB[9 / 10, 2, 0.33333333333333333333331]

LameEigenvalueB can take complex number parameters and argument:

Wolfram Language code: LameEigenvalueB[1.2 + I, 2, 0.3]
Wolfram Language code: LameEigenvalueB[1.2 + I, 2, 0.3 + I]

Evaluate LameEigenvalueB efficiently at high precision:

Wolfram Language code: LameEigenvalueB[0.9, 2, 3 / 10`500]//Timing
Wolfram Language code: LameEigenvalueB[0.9, 2, 3 / 10`500 + I]//Timing

Lists and matrices:

Wolfram Language code: LameEigenvalueB[0.9, 2, {0.1, 0.1 + I, I, 4}]
Wolfram Language code: LameEigenvalueB[0.9, 2, (| | | | :- | :---- | | π | u | | v | (π/2) |)]

Specific Values  (2)

Value of LameEigenvalueB when and :

Wolfram Language code: LameEigenvalueB[ν, 1, 0]

Value of LameEigenvalueB for and is :

Wolfram Language code: LameEigenvalueB[0, 5, 0]

For integer values of and , LameEigenvalueB is the root of a polynomial:

Wolfram Language code: LameEigenvalueB[3, 1, (1/2)]//FunctionExpand
Wolfram Language code: LameEigenvalueB[5, 2, (1/2)]//FunctionExpand

Visualization  (5)

Plot the first five LameEigenvalueB functions:

Wolfram Language code: Plot[Table[LameEigenvalueB[3 / 2, j, m], {j, 1, 5}], {m, 0, 1}, Evaluated -> True]

Plot the absolute value of the LameEigenvalueB function for complex :

Wolfram Language code: Plot[Re[LameEigenvalueB[3 / 2 + I, 3, m]], {m, 0, 1}]

Plot LameEigenvalueB as a function of its first parameter :

Wolfram Language code: Plot[{LameEigenvalueB[-1 / 2, 2, m], LameEigenvalueB[2, 2, m], LameEigenvalueB[3, 2, m]}, {m, 0, 1}]

Plot LameEigenvalueB as a function of order and elliptic parameter :

Wolfram Language code: Plot3D[LameEigenvalueB[ν, 1, m], {ν, -1 / 2, 1}, {m, 0, 1}]

Plot the family of LameEigenvalueB functions for different values of the elliptic parameter :

Wolfram Language code: Plot[Evaluate[Table[LameEigenvalueB[ν, 3, m], {m, 0, 1, 1 / 15}]], {ν, -1 / 2, 4}, PlotStyle -> Table[{Hue[i / 20], Thickness[0.002]}, {i, 20}], PlotRange -> All, Frame -> True, Axes -> False]

Series Expansions  (1)

Series expansion of LameEigenvalueB with at :

Wolfram Language code: Series[LameEigenvalueB[ν, 1, m], {m, 0, 4}]

Series expansion of LameEigenvalueB with at :

Wolfram Language code: Series[LameEigenvalueB[ν, 3, m], {m, 0, 4}]

Function Representations  (1)

TraditionalForm formatting:

Wolfram Language code: LameEigenvalueB[ν, j, m]//TraditionalForm

Applications  (1)

LameS solves the Lamé differential equation only if the parameter is specialized to LameEigenvalueB:

Wolfram Language code: y[x_] := LameS[ν, j, x, m];
Wolfram Language code: y''[x] + (LameEigenvalueB[ν, j, m] - ν(ν + 1)m JacobiSN[x, m]^2)y[x] == 0//Simplify

Properties & Relations  (2)

Use FunctionExpand to expand LameEigenvalueB for integer values of and :

Wolfram Language code: LameEigenvalueB[7, 5, (3/4)]//FunctionExpand

LameEigenvalueB satisfies a symmetry relation for integer values of and and :

Wolfram Language code: With[{n = 8, j = 5, m = 1 / 2}, LameEigenvalueB[n, j, m] + LameEigenvalueB[n, n - j + 1, 1 - m] - n(n + 1)//FunctionExpand//FullSimplify]

Possible Issues  (1)

LameEigenvalueB is not defined if is a negative integer:

Wolfram Language code: LameEigenvalueB[ν, -3, m]

LameEigenvalueB is not defined if is not an integer:

Wolfram Language code: LameEigenvalueB[ν, 3 / 2, m]
Wolfram Research (2020), LameEigenvalueB, Wolfram Language function, https://reference.wolfram.com/language/ref/LameEigenvalueB.html.

Text

Wolfram Research (2020), LameEigenvalueB, Wolfram Language function, https://reference.wolfram.com/language/ref/LameEigenvalueB.html.

CMS

Wolfram Language. 2020. "LameEigenvalueB." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LameEigenvalueB.html.

APA

Wolfram Language. (2020). LameEigenvalueB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LameEigenvalueB.html

BibTeX

@misc{reference.wolfram_2026_lameeigenvalueb, author="Wolfram Research", title="{LameEigenvalueB}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/LameEigenvalueB.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_lameeigenvalueb, organization={Wolfram Research}, title={LameEigenvalueB}, year={2020}, url={https://reference.wolfram.com/language/ref/LameEigenvalueB.html}, note=[Accessed: 12-June-2026]}