LameS

LameS[ν,j,z,m]

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

Details

  • LameS belongs to the Lamé class of functions and solves boundary-value problems for Laplace's equation in ellipsoidal and spheroconal coordinates, as well as occurring in other problems of mathematical physics and quantum mechanics.
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • LameS[ν,j,z,m] satisfies the Lamé differential equation , with the Lamé eigenvalue given by LameEigenvalueB[ν,j,m], and where TemplateBox[{z, m}, JacobiSN] is the Jacobi elliptic function JacobiSN[z,m].
  • For certain special arguments, LameS automatically evaluates to exact values.
  • LameS can be evaluated to arbitrary numerical precision for arbitrary complex argument.
  • LameS automatically threads over lists.
  • LameS[ν,j,z,0]=Sin[j(-z)].
  • LameS[ν,j,z,m] is proportional to HeunG[a,q,α,β,γ,δ,ξ], where , if the parameters of HeunG are specialized as follows: .

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot the LameS function for and :

Series expansion of LameS at the origin:

Scope  (27)

Numerical Evaluation  (5)

Evaluate to high precision:

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

LameS can take complex number parameters and argument:

Evaluate LameS efficiently at high precision:

Lists and matrices:

Specific Values  (3)

Value of LameS when and :

Value of LameS when and :

Some poles of LameS:

For integer values of and , LameS can be expressed entirely in terms of Jacobi elliptic functions:

Visualization  (6)

Plot the first three even LameS functions:

Plot the first three odd LameS functions:

Plot the absolute value of the LameS function for complex parameters:

Plot LameS as a function of its first parameter :

Plot LameS as a function of and elliptic parameter :

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

Function Properties  (2)

When is even, LameS is a periodic function of real argument with a period 2 EllipticK[m] and has an initial value LameS[ν,j,0,m]=0:

When is odd, LameS is a periodic function of real argument with a period 4 EllipticK[m]:

Differentiation  (3)

The -derivative of LameS is LameSPrime:

Higher derivatives of LameS are calculated using LameSPrime:

Derivatives of LameS for specific cases of parameters:

Integration  (3)

Indefinite integrals of LameS cannot be expressed in elementary or other special functions:

Definite numerical integrals of LameS:

More integrals with LameS:

Series Expansions  (3)

Series expansion of LameS at the origin:

Coefficient of the second term of this expansion:

Plot the first- and third-order approximations for LameS around :

Series expansion for LameS at any ordinary complex point:

Function Representations  (2)

LameS cannot be represented in terms of MeijerG:

TraditionalForm formatting:

Applications  (1)

LameS solves the Lamé differential equation when h=LameEigenvalueB[ν,j,m]:

Properties & Relations  (2)

LameS is an even function when is a positive odd integer:

LameS is an odd function when is a positive even integer:

Use FunctionExpand to expand LameS for integer values of and :

Possible Issues  (1)

LameS is not defined if is a negative integer:

LameS is not defined if is not an integer:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2023_lames, author="Wolfram Research", title="{LameS}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/LameS.html}", note=[Accessed: 06-December-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_lames, organization={Wolfram Research}, title={LameS}, year={2020}, url={https://reference.wolfram.com/language/ref/LameS.html}, note=[Accessed: 06-December-2023 ]}