# 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: