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}]