On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function

Abstract

We examine the asymptotic nature as k → ∞ of the coefficientsck(�) appearing in the uniform asymptotic expansion of the incomplete gamma function ( a,z) whereis a variable that depends on the ratio z/a. It is shown that this expansion diverges like the familiar factorial divided by a power dependence multiplied by a function fk(�). For values ofnear the real axis, fk(�) is a slowly varying function, but in the left half-plane, there are two lobes situated symmetrically about the negative realaxis in which fk(�) becomes large. The asymptotic expansion of fk(�) as k → ∞ is found to reveal a resurgence-type structure in which the high-order coefficients are related tothe low-order coeffi- cients. Numerical examples are given to illustrate the growth of the coefficients ck(�).