The differential equation of Mathieu, or “equation of the elliptic cylinder functions,”

occurs in many physical and astronomical problems. From the general theory of linear differential equations, we learn that its solution is of the type

where A and B denote arbitrary constants, μ is a constant depending on the constants a and q of the differential equation, and φ(z) and ψ(z) are periodic functions of z. For certain values of a and q the constant μ vanishes, and the solution y is then a purely periodic function of z; but in general μ is different from zero.