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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 ar for even Mathieu functions with characteristic exponent r and parameter q
MathieuCharacteristicB[r,q]characteristic value br 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 y+[a-2qcos (2z)]y=0. This equation appears in many physical situations that involve elliptical shapes or periodic potentials. The function MathieuC is defined to be even in z, while MathieuS is odd.
When q=0 the Mathieu functions are simply and . For non-zero q, the Mathieu functions are only periodic in z for certain values of a. Such Mathieu characteristic values are given by MathieuCharacteristicA[r, q] and MathieuCharacteristicB[r, q] with r an integer or rational number. These values are often denoted by ar and br.
For integer r, the even and odd Mathieu functions with characteristic values ar and br are often denoted cer (z, q) and ser (z, q), respectively. Note the reversed order of the arguments z and q.
According to Floquet's Theorem any Mathieu function can be written in the form eirzf (z), where f (z) has period 2 and r is the Mathieu characteristic exponent MathieuCharacteristicExponent[a, q]. When the characteristic exponent r is an integer or rational number, the Mathieu function is therefore periodic. In general, however, when r is not a real integer, ar and br turn out to be equal.
This shows the first five characteristic values ar as functions of q.
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