# 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:

## 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: