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Zbl 0877.34011
Dunster, T.M.
Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. (English)
[J] Proc. R. Soc. Lond., Ser. A 452, No.1949, 1331-1349 (1996). ISSN 0080-4630

The author derives asymptotic approximations for solutions of the differential equation $${d^2W\over d\xi^2}= (u^2\xi^2+ \beta u+\psi(u,\xi))W,\tag1$$ where $u$ is a large positive parameter, $\beta$ bounded (real or complex), the independent variable $\xi$ lies in some bounded or unbounded complex domain in which $\psi(u,\xi)$ is holomorphic and $o(u/\ln(u))$ uniformly as $u\to\infty$. Asymptotic approximations are constructed for solutions of (1) in terms of parabolic cylinder functions. The theory is applied to the incomplete gamma function $\Gamma(\alpha,z)$.
[S.Staněk (Olomouc)]

MSC 2000:: *34A30 Linear ODE and systems
33B15 Gamma-functions, etc.
33B20 Incomplete beta and gamma functions
34E10 Asymptotic perturbations (ODE)

Keywords: second-order linear differential equation; asymptotic approximations; parabolic cylinder functions; incomplete gamma function

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