## 3.2.13 Mathieu and Related Functions

 MathieuC[a, q, z] even Mathieu functions with characteristic value a and parameter q MathieuS[b, q, z] odd Mathieu function with characteristic value b and parameter q MathieuCPrime[a, q, z] and MathieuSPrime[b, q, z] z derivatives of Mathieu functions MathieuCharacteristicA[r, q] characteristic value for even Mathieu functions with characteristic exponent r and parameter q MathieuCharacteristicB[r, q] characteristic value for odd Mathieu functions with characteristic exponent r and parameter q MathieuCharacteristicExponent[a, q] characteristic exponent for Mathieu functions with characteristic value a and parameter q

Mathieu and related functions.

The Mathieu functions MathieuC[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}]

 Out[1]=

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