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Built-in Mathematica Symbol

MathieuCharacteristicExponent

MathieuCharacteristicExponent[a, q]
gives the characteristic exponent r for Mathieu functions with characteristic value a and parameter q.

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.

Evaluate numerically:

Evaluate for complex arguments:

Evaluate to high precision:

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

Simple exact values are generated automatically:

MathieuCharacteristicExponent 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:

This shows the stability diagram for the Mathieu equation:

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

This shows the band gaps in a periodic potential: