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On a simplified asymptotic formula for the Mathieu function of the third kind. (English) Zbl 0622.33005
This paper considers the asymptotic form of solutions of the equation y xx =(u 2 -2h 2 cosh2x)y for fixed real values of x and h and large complex values of u. Attention is focused on that solution known as the Mathieu function of the third kind, M ν (3) (x), and for values of u in the half plane Re(u)0. The basic asymptotic formulas require the determination of an elliptic integral but, when u is large, it is shown how this integral can be approximated by elementary functions.
MSC:
33E10Lamé, Mathieu, and spheroidal wave functions
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)