# MathieuCPrime

MathieuCPrime[a,q,z]

gives the derivative with respect to z of the even Mathieu function with characteristic value a and parameter q.

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• For certain special arguments, MathieuCPrime automatically evaluates to exact values.
• MathieuCPrime can be evaluated to arbitrary numerical precision.
• MathieuCPrime automatically threads over lists.

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

## Scope(18)

### Numerical Evaluation(4)

Evaluate numerically to high precision:

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

Evaluate for complex arguments and parameters:

Evaluate MathieuCPrime efficiently at high precision:

### Specific Values(3)

Simple exact values are generated automatically:

Find a zero of MathieuCPrime:

MathieuCPrime is an odd function:

### Visualization(2)

Plot the MathieuCPrime function:

Plot the real part of MathieuCPrime for and :

Plot the imaginary part of MathieuCPrime for and :

### Function Properties(4)

MathieuCPrime has singularities and discontinuities when the characteristic exponent is an integer:

is neither nondecreasing nor nonincreasing:

MathieuCPrime is neither non-negative nor non-positive:

MathieuCPrime is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for and :

Plot higher derivatives for and :

MathieuCPrime is the derivative of MathieuC:

### Series Expansions(2)

Taylor expansion:

Plot the first three approximations for MathieuCPrime around :

Taylor expansion of MathieuCPrime at a generic point:

## Applications(1)

Mathieu functions arise as solutions of the Laplace equation in an ellipse:

This defines the square of the gradient (the local kinetic energy of a vibrating membrane):

This finds a zero:

This plots the absolute value of the gradient of an eigenfunction:

## Neat Examples(1)

Phase space plots of the Mathieu function:

Wolfram Research (1996), MathieuCPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuCPrime.html.

#### Text

Wolfram Research (1996), MathieuCPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuCPrime.html.

#### CMS

Wolfram Language. 1996. "MathieuCPrime." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MathieuCPrime.html.

#### APA

Wolfram Language. (1996). MathieuCPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MathieuCPrime.html

#### BibTeX

@misc{reference.wolfram_2023_mathieucprime, author="Wolfram Research", title="{MathieuCPrime}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuCPrime.html}", note=[Accessed: 17-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_mathieucprime, organization={Wolfram Research}, title={MathieuCPrime}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuCPrime.html}, note=[Accessed: 17-April-2024 ]}