LameCPrime

LameCPrime[ν,j,z,m]

gives the -derivative of the ^(th) Lamé function TemplateBox[{nu, j, z, m}, LameC] of order with elliptic parameter .

Details

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

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot the LameCPrime function for and :

Series expansion of LameCPrime at the origin:

Scope  (26)

Numerical Evaluation  (5)

Evaluate to high precision:

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

LameCPrime can take complex number parameters and argument:

Evaluate LameCPrime efficiently at high precision:

Lists and matrices:

Specific Values  (3)

Value of LameCPrime when and :

Value of LameCPrime when and :

Some poles of LameCPrime:

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

Visualization  (6)

Plot the first three even LameCPrime functions:

Plot the first three odd LameCPrime functions:

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

Plot LameCPrime as a function of its first parameter :

Plot LameCPrime as a function of and elliptic parameter :

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

Function Properties  (2)

When is even, LameCPrime is a periodic function of real argument with a period 2EllipticK[m] and has an initial value LameCPrime[ν,j,0,m]=0:

When is odd, LameCPrime is a periodic function of real argument with a period 4EllipticK[m]:

Differentiation  (2)

The -derivative of LameCPrime involves LameC function:

Derivatives of LameCPrime for specific cases of parameters:

Integration  (3)

Indefinite integral of LameCPrime is LameC:

Definite numerical integrals of LameCPrime:

More integrals with LameCPrime:

Series Expansions  (3)

Series expansion of LameCPrime at the origin:

Coefficient of the second term of this expansion:

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

Series expansion for LameCPrime at any ordinary complex point:

Function Representations  (2)

LameCPrime cannot be represented in terms of MeijerG:

TraditionalForm formatting:

Applications  (1)

Use the LameCPrime function to calculate the derivatives of LameC:

Properties & Relations  (2)

LameCPrime is an even function when is a positive odd integer:

LameCPrime is an odd function when is a non-negative even integer:

Use FunctionExpand to expand LameCPrime for integer values of and :

Possible Issues  (1)

LameCPrime is not defined if is a negative integer:

LameCPrime is not defined if is not an integer:

Wolfram Research (2020), LameCPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/LameCPrime.html.

Text

Wolfram Research (2020), LameCPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/LameCPrime.html.

CMS

Wolfram Language. 2020. "LameCPrime." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LameCPrime.html.

APA

Wolfram Language. (2020). LameCPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LameCPrime.html

BibTeX

@misc{reference.wolfram_2024_lamecprime, author="Wolfram Research", title="{LameCPrime}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/LameCPrime.html}", note=[Accessed: 25-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_lamecprime, organization={Wolfram Research}, title={LameCPrime}, year={2020}, url={https://reference.wolfram.com/language/ref/LameCPrime.html}, note=[Accessed: 25-April-2024 ]}