# MathieuCharacteristicExponent

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

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• All Mathieu functions have the form where has period and r is the Mathieu characteristic exponent.
• For certain special arguments, MathieuCharacteristicExponent automatically evaluates to exact values.
• MathieuCharacteristicExponent can be evaluated to arbitrary numerical precision.
• MathieuCharacteristicExponent 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(13)

### Numerical Evaluation(5)

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 a value of q for which MathieuCharacteristicExponent[3,q]=1.7:

### Visualization(3)

Plot the MathieuCharacteristicExponent function for integer parameters:

Plot the MathieuCharacteristicExponent function for noninteger parameters:

Plot the real part of MathieuCharacteristicExponent:

Plot the imaginary part of MathieuCharacteristicExponent:

### Function Properties(3)

is neither non-decreasing nor non-increasing:

is neither non-negative nor non-positive:

is neither convex nor concave:

## Applications(2)

Solve the Schrödinger equation with periodic potential:

By the Bloch theorem, solutions are bounded provided is within an energy band. The energy gap corresponds to a range of where MathieuCharacteristicExponent has a non-vanishing imaginary part:

This shows the stability diagram for the Mathieu equation:

## Properties & Relations(1)

The characteristic exponent and the characteristic are inverses of each other:

## Neat Examples(1)

This shows the band gaps in a periodic potential: