Uniform asymptotic approximation of Mathieu functions (Q1902352)

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 \(- 2q \leq a \leq (2 - d)q\), 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.