MathieuCharacteristicExponent

MathieuCharacteristicExponent[a,q]

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:

MathieuCharacteristicExponent threads elementwise over lists:

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)

MathieuCharacteristicExponent[3,x] is neither non-decreasing nor non-increasing:

MathieuCharacteristicExponent[3,x] is neither non-negative nor non-positive:

MathieuCharacteristicExponent[3,x] 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:

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MathieuCharacteristicExponent

Details

Examples

Basic Examples (3)