On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function
type: Item
instance of: scholarly article
title: On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function
zbMATH Open document ID: 0921.33001
author: K. Appert
published in: Methods and Applications of Analysis
publication date: 1999-09-24T00:00:00Z
review text: The authors study coefficients \(c_k(\eta)\) in the asymptotic expansion of the normalized incomplete gamma function \(Q(a,z)\equiv \Gamma(a,z)/\Gamma(a)\) as \(a\rightarrow\infty\), as given by \textit{N. M. Temme} [SIAM J. Math. Anal. 10, 757-766 (1979; Zbl 0412.33001) and ``Special functions'' (1996; Zbl 0856.33001)]: \[ Q(a,z)\sim {1\over 2}\text{erfc} \Biggl( \eta\sqrt{{1\over 2}a}\Biggr) + {e^{-a\eta^2/2}\over \sqrt{2\pi a}}\sum_{k=0}^{\infty} c_k(\eta)a^{-k} \] where \[ \eta=\{2(\mu-\log{(1+\mu)}\}^{1/2},\qquad \mu=\lambda-1,\;\lambda={z\over a}. \] First the asymptotic behavior of \(c_k(\eta)\) as \(k\rightarrow\infty\) is given (showing a different behavior on the left and right \(\eta\) half plane) using the saddle point method on the two integrals appearing in an explicit expression for these coefficients. The asymptotic behavior is also derived using the MacLaurin expansion of the \(c_k\)'s. Finally some numerical results are discussed, showing a.o. the possibility to use the coefficients \(c_k(\eta)\) for optimal truncation, needed to depict the accuracy obtained by the asymptotics for \(Q(a,z)\) for fixed \(| a| \) and \(| z| \). A typical example of hard analysis.
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 33B20
Mathematics Subject Classification ID: 30E15
zbMATH DE Number: 1292477
zbMATH Keywords: incomplete gamma function
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: resurgence
MaRDI profile type: Publication
DOI: 10.4310/MAA.1998.V5.N3.A1
Recommended article: A uniform asymptotic expansion for the incomplete gamma function
Recommended article: Approximations for the higher order coefficients in an asymptotic expansion for the gamma function
Recommended article: Asymptotic expansions for the incomplete gamma functions
Recommended article: Asymptotic expansions for generalized gamma and incomplete gamma functions
Recommended article: On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function
Recommended article: Asymptotic expansions for the incomplete gamma function in the transition regions
Recommended article: Error bounds for the uniform asymptotic expansion of the incomplete gamma function
Recommended article: The asymptotic expansion of a generalised incomplete gamma function.
Recommended article: Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters
Recommended article: Asymptotics and closed form of a generalized incomplete gamma function
description: scientific article; zbMATH DE number 1292477