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LameC

LameC[ν,j,z,m]

gives the Lamé function of order with elliptic parameter .

Details

LameC 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.
LameC[ν,j,z,m] satisfies the Lamé differential equation , with the Lamé eigenvalue given by LameEigenvalueA[ν,j,m], and where is the Jacobi elliptic function JacobiSN[z,m].
For certain special arguments, LameC automatically evaluates to exact values.
LameC can be evaluated to arbitrary numerical precision for an arbitrary complex argument.
LameC automatically threads over lists.
LameC[ν,0,z,0]= and LameC[ν,j,z,0]=Cos[j(-z)].
LameC[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 LameC function for and :

Series expansion of LameC at the origin:

Scope (27)

Numerical Evaluation (5)

Evaluate to high precision:

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

LameC can take complex number parameters and argument:

Evaluate LameC efficiently at high precision:

Lists and matrices:

Specific Values (3)

Value of LameC when and :

Value of LameC when and :

Some poles of LameC:

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

Visualization (6)

Plot the first three even LameC functions:

Plot the first three odd LameC functions:

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

Plot LameC as a function of its first parameter :

Plot LameC as a function of and elliptic parameter :

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

Function Properties (2)

When is even, LameC is a periodic function of real argument with a period 2EllipticK[m]:

When is odd, LameC is a periodic function of real argument with a period 4EllipticK[m] and has an initial value LameC[ν,j,0,m]=0:

Differentiation (3)

The -derivative of LameC is LameCPrime:

Higher derivatives of LameC are calculated using LameCPrime:

Derivatives of LameC for specific cases of parameters:

Integration (3)

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

Definite numerical integrals of LameC:

More integrals with LameC:

Series Expansions (3)

Series expansion of LameC at the origin:

Coefficient of the second-order term of this expansion:

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

Series expansion for LameC at any ordinary complex point:

Function Representations (2)

LameC cannot be represented in terms of MeijerG:

TraditionalForm formatting:

Applications (1)

LameC solves the Lamé differential equation when h=LameEigenvalueA[ν,j,m]:

Properties & Relations (2)

LameC is an even function when is a non-negative even integer:

LameC is an odd function when is a positive odd integer:

Use FunctionExpand to expand LameC for integer values of and :

Possible Issues (1)

LameC is not defined if is a negative integer:

LameC is not defined if is not an integer:

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