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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 $w'' = \{2q \cos (2z) - a\} w$ for $z$ complex, and $q$ and $a$ real, $q \to \infty$. The approximations are uniformly valid for $- 2q \le a \le (2 - d)q$, 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.
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)