# 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

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## 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:

## 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: