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

*(English)*Zbl 0877.34011The author derives asymptotic approximations for solutions of the differential equation

$$\frac{{d}^{2}W}{d{\xi}^{2}}=({u}^{2}{\xi}^{2}+\beta u+\psi (u,\xi ))W,\phantom{\rule{2.em}{0ex}}\left(1\right)$$

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 $o(u/ln(u\left)\right)$ 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 ${\Gamma}(\alpha ,z)$.

Reviewer: S.Staněk (Olomouc)

##### MSC:

34A30 | Linear ODE and systems, general |

33B15 | Gamma, beta and polygamma functions |

33B20 | Incomplete beta and gamma functions |

34E10 | Perturbations, asymptotics (ODE) |