Dunster, T. M. Error analysis in a uniform asymptotic expansion for the generalised exponential integral. (English) Zbl 0870.33002 J. Comput. Appl. Math. 80, No. 1, 127-161 (1997). 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 \(0\leq \arg (z/p)\leq 2\pi\). 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. Cited in 4 Documents MSC: 33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 34E05 Asymptotic expansions of solutions to ordinary differential equations 30E10 Approximation in the complex plane 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) Keywords:uniform asymptotic expansions; incomplete gamma functions; turning point theory; error function; generalised exponential integral PDFBibTeX XMLCite \textit{T. M. Dunster}, J. Comput. Appl. Math. 80, No. 1, 127--161 (1997; Zbl 0870.33002) Full Text: DOI Digital Library of Mathematical Functions: §2.11(iii) Exponentially-Improved Expansions ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations §8.20(ii) Large 𝑝 ‣ §8.20 Asymptotic Expansions of 𝐸_𝑝(𝑧) ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions References: [1] Berry, M. V., Uniform asymptotic smoothing of Stokes’ discontinuities, (Proc. Roy. Soc. Lond. Ser. A, 422 (1989)), 7-21 · Zbl 0683.33004 [2] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, On Lambert’s \(W\); R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, On Lambert’s \(W\) · Zbl 0863.65008 [3] Dunster, T. M., Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete Gamma function, (Proc. Roy. Soc. London Ser. A, 452 (1996)), 1331-1349 · Zbl 0877.34011 [4] Dunster, T. M., Asymptotics of the generalised exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes’ discontinuities, (Proc. Roy. Soc. London Ser. A, 452 (1996)), 1351-1367 · Zbl 0866.41025 [5] Dunster, T. M., Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one, Methods Appl. Anal., 3, 1, 109-134 (1996) · Zbl 0859.34039 [6] A.B. Olde Daalhuis, Hyperterminants I, J. Comput. Appl. Math, to appear.; A.B. Olde Daalhuis, Hyperterminants I, J. Comput. Appl. Math, to appear. [7] Olde Daalhuis, A. B.; Olver, F. W.J., Exponentially-improved asymptotic solutions of ordinary differential equations II: irregular singularities of rank one, (Proc. Roy. Soc. Lond. Ser. A, 445 (1994)), 39-56 · Zbl 0809.34007 [8] Olver, F. W.J., Asymptotics and Special Functions (1974), Academic Press: Academic Press New York, reprinted by A.K. Peters, Wellesley, Massachusetts, 1997 · Zbl 0303.41035 [9] Olver, F. W.J., Uniform, exponentially-improved asymptotic expansions for the generalised exponential integral, SIAM J. Math. Anal., 22, 1460-1489 (1991) · Zbl 0743.41025 [10] Temme, N. M., The asymptotic expansion of the incomplete Gamma functions, SIAM J. Math. Anal., 10, 757-766 (1979) · Zbl 0412.33001 [11] Temme, N. M., Uniform asymptotics for the incomplete Gamma functions starting from negative values of the parameters, Methods Appl. Anal. (1996), to appear · Zbl 0863.33002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.