THE CONVERGING FACTOR FOR THE EXPONENTIAL INTEGRAL,

The term converging factor, is generally defined as the factor by which the final term of a truncated series must be multiplied to yield the remainder of the series. In this report the converging factor associated with the asymptotic series for the exponential integral Eix of both positive and negative real argument x is discussed in detail, and numerical values therof for integral arguments are tabulated to 45 or more decimal places. Auxiliary tables are presented to permit the evaluation of this factor to comparable accuracy for intermediate values of the argument. Asymptotic series for the converging factor are rigorously developed, and the exact rational values of the first 21 coefficients are presented. As a byproduct, the first 20 nontrivial coefficients of Stirlings asymptotic series for the factorial function are deduced. A method for evaluating the exponential integral is presented in detail, and original tables of values of the exponential integral are given to 44 significant figures for integral values of x extending from 5 through 20 and to 50 decimal places for integral values of x ranging from -5 to -20, inclusive. Author