Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how

LameSPrime

LameSPrime[ν,j,z,m]

gives the -derivative of the Lamé function of order with elliptic parameter .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
LameSPrime belongs to the Lamé class of functions.
For certain special arguments, LameSPrime automatically evaluates to exact values.
LameSPrime can be evaluated to arbitrary numerical precision for an arbitrary complex argument.
LameSPrime automatically threads over lists.
LameSPrime[ν,j,z,0]=-j Cos[j(-z)].

Examples

open allclose all

Basic Examples (3)

Evaluate numerically:

Plot the LameSPrime function for and :

Series expansion of LameSPrime at the origin:

Scope (26)

Numerical Evaluation (5)

Evaluate to high precision:

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

LameSPrime can take complex number parameters and argument:

Evaluate LameSPrime efficiently at high precision:

Lists and matrices:

Specific Values (3)

Value of LameSPrime when and :

Value of LameSPrime when and :

Some poles of LameSPrime:

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

Visualization (6)

Plot the first three even LameSPrime functions:

Plot the first three odd LameSPrime functions:

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

Plot LameSPrime as a function of its first parameter :

Plot LameSPrime as a function of and elliptic parameter :

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

Function Properties (2)

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

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

Differentiation (2)

The -derivative of LameSPrime involves the LameS function:

Derivatives of LameSPrime for specific cases of parameters:

Integration (3)

Indefinite integral of LameSPrime is LameS:

Definite numerical integrals of LameSPrime:

More integrals with LameSPrime:

Series Expansions (3)

Series expansion of LameSPrime at the origin:

Coefficient of the second term of this expansion:

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

Series expansion for LameSPrime at any ordinary complex point:

Function Representations (2)

LameSPrime cannot be represented in terms of MeijerG:

TraditionalForm formatting:

Applications (1)

Use the LameSPrime function to calculate the derivatives of LameS:

Properties & Relations (2)

LameSPrime is an even function when is an positive even integer:

LameSPrime is an odd function when is an positive odd integer:

Use FunctionExpand to expand LameSPrime for integer values of and :

Possible Issues (1)

LameSPrime is not defined if is a negative integer:

LameSPrime is not defined if is not an integer:

Top