# 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

open allclose all

## 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(14)

### 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 MathieuSPrime efficiently at high precision:

### 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 :

### 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:

Introduced in 1996
(3.0)