On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function
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.
scientific article; zbMATH DE number 1292477