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
open allclose allBasic Examples (3)
Scope (13)
Numerical Evaluation (5)
MathieuCharacteristicExponent threads elementwise over lists:
The precision of the output tracks the precision of the input:
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:
Properties & Relations (2)
The characteristic exponent and the characteristic are inverses of each other:
From the plot, you can see that MathieuCharacteristicExponent[x,0]=
:
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