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Uniform asymptotic approximation of Mathieu functions. (English) Zbl 0835.33014
Uniform asymptotic approximations are derived for solutions of Mathieu’s equation w '' ={2qcos(2z)-a}w for z complex, and q and a real, q. The approximations are uniformly valid for -2qa(2-d)q, where d is an arbitrarily small positive constant. The approximations involve both elementary functions and Whittaker functions. Error bounds are included or available for all approximations. The paper also gives an introduction to well-known basic properties of Mathieu’s equation and its solutions, which are relevant to the paper.
MSC:
33E10Lamé, Mathieu, and spheroidal wave functions
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)