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Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. (English) Zbl 0877.34011

The author derives asymptotic approximations for solutions of the differential equation

d 2 W dξ 2 =(u 2 ξ 2 +βu+ψ(u,ξ))W,(1)

where u is a large positive parameter, β bounded (real or complex), the independent variable ξ lies in some bounded or unbounded complex domain in which ψ(u,ξ) is holomorphic and o(u/ln(u)) uniformly as u. Asymptotic approximations are constructed for solutions of (1) in terms of parabolic cylinder functions. The theory is applied to the incomplete gamma function Γ(α,z).

MSC:
34A30Linear ODE and systems, general
33B15Gamma, beta and polygamma functions
33B20Incomplete beta and gamma functions
34E10Perturbations, asymptotics (ODE)