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 a_r and b_r.

For integer r, the even and odd Mathieu functions with characteristic values a_r and b_r are often denoted ce_r (z, q) and se_r (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 e^irzf (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, a_r and b_r turn out to be equal.