The author reconsiders the classical Mathieu equation
in the restricted case where it has nontrivial -periodic or -antiperiodic solutions. In the first case it holds while in the second case , yielding as a function of , say (eigenvalues). The aim is to find a new, sharper estimation for the radius of convergence of the power series of about the point . The result, which seems to be the best one, is as follows:
Here, refers the radius of convergence of power series of while denotes the complete elliptic integral of the first kind and . As to the modulus , it is determined through the relation , being the corresponding complete elliptic integral of the second kind.