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MathieuCharacteristicExponent

MathieuCharacteristicExponent
gives the characteristic exponent r for Mathieu functions with characteristic value a and parameter q.
  • 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.
Evaluate numerically:
Evaluate numerically:
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Click for copyable input
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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:
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