MathieuCharacteristicA[r, q] gives the characteristic value for even Mathieu functions with characteristic exponent r and parameter q.
Mathematical function (see Section A.3.10).
The characteristic value gives the value of the parameter in for which the solution has the form where is an even function of with period .
See The Mathematica Book: Section 3.2.12.
See also: MathieuCharacteristicB.