|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|
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 .
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.