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On simplified asymptotic formulas for a class of Mathieu functions. (English) Zbl 0557.33005
Author’s abstract. This paper considers the asymptotic form of solutions of the equation y xx =(u 2 +2h 2 cosh2x)y for real values of x and h and large values of u. Attention is focussed on the solution ψ (x,u) that tends to zero as x and for values of u in the half plane Re(u)0. The basic asymptotic formulas that appear require the determination of an elliptic integral but, when u is large, it is shown how this integral can be suitably approximated by elementary functions. An asymptotic formula is derived which gives the large zeros of the function ψ (x,u) regarded as a function of u, the quantity x being supposed prescribed and positive.
Reviewer: P.A.McCoy
MSC:
33E10Lamé, Mathieu, and spheroidal wave functions