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LameS

LameS[ν,j,z,m]

gives the Lamé function 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 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:

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