# MathieuSPrime

MathieuSPrime[a,q,z]

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

# Details

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

Series expansion at the origin:

## Scope(19)

### Numerical Evaluation(5)

Evaluate numerically to high precision:

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

Evaluate for complex arguments and parameters:

Evaluate MathieuSPrime efficiently at high precision:

Compute the elementwise values of an array:

Or compute the matrix MathieuSPrime function using MatrixFunction:

### Specific Values(3)

Simple exact values are generated automatically:

Find a zero of MathieuSPrime:

MathieuSPrime is an even function:

### Visualization(2)

Plot the MathieuSPrime function:

Plot the real part of MathieuSPrime for and :

Plot the imaginary part of MathieuSPrime for and :

### Function Properties(4)

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

is neither nondecreasing nor nonincreasing:

MathieuSPrime is neither non-negative nor non-positive:

MathieuSPrime is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for and :

Plot higher derivatives for and :

MathieuSPrime is the derivative of MathieuS:

### Series Expansions(2)

Taylor expansion:

Plot the first three approximations for MathieuSPrime around :

Taylor expansion of MathieuSPrime 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), MathieuSPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuSPrime.html.

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_mathieusprime, author="Wolfram Research", title="{MathieuSPrime}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuSPrime.html}", note=[Accessed: 11-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_mathieusprime, organization={Wolfram Research}, title={MathieuSPrime}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuSPrime.html}, note=[Accessed: 11-September-2024 ]}