# MathieuCharacteristicA

gives the characteristic value for even Mathieu functions with characteristic exponent r and parameter q.

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The characteristic value gives the value of the parameter in for which the solution has the form , where is an even function of with period .
• For certain special arguments, MathieuCharacteristicA automatically evaluates to exact values.
• MathieuCharacteristicA can be evaluated to arbitrary numerical precision.
• MathieuCharacteristicA automatically threads over lists.

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

## Scope(16)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

### Specific Values(2)

Simple exact values are generated automatically:

Find the positive maximum of :

### Visualization(3)

Plot the MathieuCharacteristicA function for integer parameters:

Plot the MathieuCharacteristicA function for noninteger parameters:

Plot the real part of MathieuCharacteristicA:

Plot the imaginary part of MathieuCharacteristicA:

### Function Properties(7)

The real domain of MathieuCharacteristicA:

Approximate function range of : is a continuous function of : is neither non-increasing nor non-decreasing: is not injective:

## Applications(3)

Symmetric periodic solutions of the Mathieu differential equation:

This shows the stability diagram for the Mathieu equation:

As a function of the first argument, MathieuCharacteristicA is a piecewise continuous function (called bands and band gaps in solid state physics):

## Neat Examples(1)

Branch points of the Mathieu characteristic along the imaginary q axis: