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Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. (English) Zbl 0859.34039

The author investigates solutions of the general homogeneous linear second order differential equation of the form

$\frac{{d}^{2}w}{d{z}^{2}}+f\left(z\right)\frac{dw}{dz}+g\left(z\right)w=0·\phantom{\rule{2.em}{0ex}}\left(*\right)$

In §2, the author gives error bounds which are uniformly valid for $0\le |\text{arg}\left(z{e}^{-\pi \left(j-1\right)i}\right)|\le 2\pi$. In §3, he gives details of the proof of the derivation of these bounds, which uses the technique of successive approximations. In §4, the author generalizes the results of §§2-3 to give exponentially improved expansions with an improved relative error term of $O\left({z}^{-m}\right)$ as $z\to \infty$ where $m$ is a prescribed fixed positive integer. In §5, he gives brief details on the extension of the error analysis to sectors, in conjunction with the results of §4. In §6, the author examines in more detail the asymptotic nature of the error bounds, which involve so-called weight functions. Finally, in §7, he gives a numerical example on the calculation of certain constants which appear.

##### MSC:
 34E05 Asymptotic expansions (ODE) 34M99 Differential equations in the complex domain 41A30 Approximation by other special function classes 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 30E10 Approximation in the complex domain 30E15 Asymptotic representations in the complex domain