Concerning Two Series for the Gamma Function

series for In r(z) contains only odd powers of z-', whereas the corresponding series for r(z) contains all powers of z-, nevertheless the latter provides an effective computational tool for the direct evaluation of r(z), especially by means of modern digital computers. For that reason, the exact (rational) values of the first twenty coefficients of Stirling's asymptotic series for 1(z) have been calculated and are tabulated herein. The second series here considered is the power series for the entire function 1/r(z). The first extensive calculation of the coefficients of this series appears to have been performed by Bourguet [2]. His 16D approximations were subsequently recalculated and corrected by Isaacson and Salzer [3]. These emended values have been reproduced in Davis [4] and in the NBS Handbook [5]. In the course of checking [6] these corrected values the present author has now recalculated these coefficients anew and extended the approximations to 31D. These new data are also tabulated in this paper, and their application is illustrated through the evaluation of the main minimum of 17(x) to 31D.