# MathieuCharacteristicB

gives the characteristic value for odd 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 odd function of with period .
• When r is not a real integer, MathieuCharacteristicB gives the same results as MathieuCharacteristicA.
• For certain special arguments, MathieuCharacteristicB automatically evaluates to exact values.
• MathieuCharacteristicB can be evaluated to arbitrary numerical precision.
• MathieuCharacteristicB 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(11)

### 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 MathieuCharacteristicB function for integer parameters:

Plot the MathieuCharacteristicB function for noninteger parameters:

Plot the real part of MathieuCharacteristicB:

Plot the imaginary part of MathieuCharacteristicB:

### Function Properties(2)

The real domain of MathieuCharacteristicB:

## 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, MathieuCharacteristicB is a piecewise continuous function (called bands and band gaps in solid-state physics):

## Possible Issues(1)

There is no zero-order MathieuCharacteristicB: ## Neat Examples(1)

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

Introduced in 1996
(3.0)