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3.2.12 Mathieu and Related Functions
Mathieu and related functions.
The Mathieu functionsMathieuC[a,q,z] and MathieuS[a,q,z] are solutions to the equation  . This equation appears in many physical situations that involve elliptical shapes or periodic potentials. The function MathieuC is defined to be even in  , while MathieuS is odd.
When  the Mathieu functions are simply  and  . For non-zero  , the Mathieu functions are only periodic in  for certain values of  . Such Mathieu characteristic values are given by MathieuCharacteristicA[
r,q] and MathieuCharacteristicB[r,q] with  an integer or rational number. These values are often denoted by  and  .
For integer  , the even and odd Mathieu functions with characteristic values  and  are often denoted  and  , respectively. Note the reversed order of the arguments  and  .
According to Floquet's Theorem any Mathieu function can be written in the form  , where  has period  and  is the Mathieu characteristic exponent
MathieuCharacteristicExponent[a,q]. When the characteristic exponent  is an integer or rational number, the Mathieu function is therefore periodic. In general, however, when  is not a real integer,  and 
turn out to be equal.
This shows the first five characteristic values  as functions of 
.
In[1]:= Plot[Evaluate[Table[MathieuCharacteristicA[r, q],
{r, 0, 4}]], {q, 0, 15}]
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