Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function

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Uniform asymptotic expansions are derived for solutions of the differential equation d 2 W/dζ 2 = ( u 2 ζ 2 + βu + ψ( u , ζ))W, which are uniformly valid for u real and large, β bounded (real or complex), and £ lying in a well-defined bounded or unbounded complex domain, which contains the origin. The function ψ(u, ζ) is assumed to be holomorphic in this domain, and is o ( u / ln( u )) uniformly as u —> oo. The approximations involve parabolic cylinder functions, and include explicit and realistic error bounds. The new theory is then applied to the complementary incomplete gamma function Γ( α , α x ), furnishing an asymptotic approximation which is uniformly valid for x lying in a complex domain which properly contains all x satisfying |arg(x)| oo.