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:

Plot the LameEigenvalueB function:

Scope  (14)

Numerical Evaluation  (5)

Evaluate to high precision:

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

LameEigenvalueB can take complex number parameters and argument:

Evaluate LameEigenvalueB efficiently at high precision:

Lists and matrices:

Specific Values  (2)

Value of LameEigenvalueB when and :

Value of LameEigenvalueB for and is :

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

Visualization  (5)

Plot the first five LameEigenvalueB functions:

Plot the absolute value of the LameEigenvalueB function for complex :

Plot LameEigenvalueB as a function of its first parameter :

Plot LameEigenvalueB as a function of order and elliptic parameter :

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

Series Expansions  (1)

Series expansion of LameEigenvalueB with at :

Series expansion of LameEigenvalueB with at :

Function Representations  (1)

TraditionalForm formatting:

Applications  (1)

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

Properties & Relations  (2)

Use FunctionExpand to expand LameEigenvalueB for integer values of and :

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

Possible Issues  (1)

LameEigenvalueB is not defined if is a negative integer:

LameEigenvalueB is not defined if is not an integer:

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_2024_lameeigenvalueb, author="Wolfram Research", title="{LameEigenvalueB}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/LameEigenvalueB.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_lameeigenvalueb, organization={Wolfram Research}, title={LameEigenvalueB}, year={2020}, url={https://reference.wolfram.com/language/ref/LameEigenvalueB.html}, note=[Accessed: 21-December-2024 ]}