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On the growth of convergence radii for the eigenvalues of the Mathieu equation. (English) Zbl 0912.34027

The author reconsiders the classical Mathieu equation

y '' +(μ+2λcos2x)y=0

in the restricted case where it has nontrivial π-periodic or π-antiperiodic solutions. In the first case it holds y(x+π)=y(x) while in the second case y(x+π)=-y(x), yielding μ as a function of λ, say μ=μ 0 (λ) (n=1,2,) (eigenvalues). The aim is to find a new, sharper estimation for the radius of convergence of the power series of μ n (λ) about the point λ=0. The result, which seems to be the best one, is as follows:

lim n infρ n n 2 kk ' K 2 =2·041834·

Here, ρ n refers the radius of convergence of power series of μ 0 (λ) while K=K(k) denotes the complete elliptic integral of the first kind and k ' =(1-k 2 ) 1/2 . As to the modulus k, it is determined through the relation 2E=K, E being the corresponding complete elliptic integral of the second kind.

MSC:
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
33E10Lamé, Mathieu, and spheroidal wave functions