# MathieuS

MathieuS[a,q,z]

gives the odd Mathieu function with characteristic value a and parameter q.

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The Mathieu functions satisfy the equation .
• For certain special arguments, MathieuS automatically evaluates to exact values.
• MathieuS can be evaluated to arbitrary numerical precision.
• MathieuS 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:

Series expansion at the origin:

## Scope(19)

### 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 MathieuS efficiently at high precision:

### Specific Values(3)

Simple exact values are generated automatically:

Find a local maximum as the root of in the maximum's neighborhood:

MathieuS is an odd function:

### Visualization(3)

Plot the MathieuS function:

Plot the real part of MathieuS for and :

Plot the imaginary part of MathieuS for and :

Plot the real part of MathieuS for and :

Plot the imaginary part of MathieuS for and :

### Function Properties(4)

MathieuS has singularities and discontinuities when the characteristic exponent is an integer: is neither nondecreasing nor nonincreasing:

MathieuS is neither non-negative nor non-positive:

MathieuS is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for and :

Plot higher derivatives for and :

Mathieu functions are the solutions to the differential equation :

### Series Expansions(2)

Taylor expansion:

Plot the first three approximations for MathieuS around :

Taylor expansion of MathieuS at a generic point:

## 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:

Solve the Laplace equation in an ellipse using separation of variables:

This finds a zero:

This plots an eigenfunction. It vanishes at the ellipse boundary:

## Possible Issues(1)

Machine-precision input is insufficient to give a correct answer:

## Neat Examples(1)

Phase space plots of the Mathieu function: