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Zbl 0870.33002
Dunster, T.M.
Error analysis in a uniform asymptotic expansion for the generalised exponential integral. (English)
[J] J. Comput. Appl. Math. 80, No.1, 127-161 (1997). ISSN 0377-0427

Summary: Uniform asymptotic expansions are derived for the generalised exponential integral $E_p(z)$, where both $p$ and $z$ are complex. These are derived by examining the differential equation satisfied by $E_p(z)$, an equation which possesses a double turning point at $z/p= -1$. The expansions, which involve the complementary error function, together approximate $E_p(z)$ as $|p|\to\infty$, uniformly for all non-zero complex $z$ satisfying $0\le \arg (z/p)\le 2\pi$. The error terms associated with the truncated expansions are shown to be solutions of inhomogeneous differential equations, and from these explicit and realistic bounds are derived. By employing the maximum-modulus theorem the bounds are then simplified to make them more conductive to numerical evaluation.

MSC 2000:: *33B20 Incomplete beta and gamma functions
34E20 Asymptotic singular perturbations, methods (ODE)
34E05 Asymptotic expansions (ODE)
30E10 Approximation in the complex domain
41A60 Asymptotic problems in approximation

Keywords: uniform asymptotic expansions; incomplete gamma functions; turning point theory; error function; generalised exponential integral

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