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Mathieu and Related Functions

MathieuC[a,q,z]even Mathieu functions with characteristic value a and parameter q
MathieuS[b,q,z]odd Mathieu functions 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 r for Mathieu functions with characteristic value a and parameter q

Mathieu and related functions.

The Mathieu functions MathieuC and MathieuS 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 nonzero , the Mathieu functions are only periodic in for certain values of . Such Mathieu characteristic values are given by MathieuCharacteristicA and MathieuCharacteristicB 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. 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 .
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