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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}]