MathieuS

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

For certain special arguments, MathieuS automatically evaluates to exact values.

Evaluate numerically:

Out[1]=

Out[1]=

Evaluate for complex arguments and parameters:

Evaluate to high precision:

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

Simple exact values are generated automatically:

MathieuS threads element-wise over lists:

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:

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

Phase space plots of the Mathieu function: