Gevrey asymptotics and Stieltjes transforms of algebraically decaying functions

The development of asymptotic expansions of Stieltjes transforms of exponentially decaying functions has been well established. In this paper, we are concerned with the more difficult case in which the functions decay only algebraically at infinity. By using a Gevrey-type condition, we obtain an exponentially improved asymptotic expansion, and give three representation theorems to show that the Stieltjes transform of algebraically decaying functions can be written as the difference of two integral transforms with exponentially decaying kernels, thus making the asymptotic theory developed for integral transforms with exponentially decaying kernels relevant to Stieltjes transforms of algebraically decaying functions, including the smoothing of the Stokes phenomenon.