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] andMathieuSPrime[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[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 nonzero , 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 .
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