04-23 Armando G. M. Neves

Upper and lower bounds on Mathieu characteristic numbers of integer orders (278K, PDF) Feb 2, 04

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Abstract. Consider Mathieu's equation ${d^2y \over dt^2}\,+\, (a- 2q \cos 2t) \, y \,=\,0$, $a, q, t \in \mathbb{R}$. The Mathieu characteristic numbers of integer orders (MCNs) are, for each real value for $q$, the values of $a$ such that the equation has a periodic solution of period 2$\pi$. For each MCN we construct sequences of upper and lower bounds both converging to the MCN. The bounds arise as zeros of polynomials in sequences generated by recursion. This result is based on a constructive proof of convergence for Ince's continued fractions. An important role is also played by the fact that the continued fractions define meromorphic functions.

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