Compact Analytical Fit to the Exponential Integral E1(x)

摘要

A four-parameter fit is developed for the class of integrals known as the exponential integral (real branch). Unlike other fits that are piecewise in nature, the current fit to the exponential integral is valid over the complete domain of the function (compact) and is everywhere accurate to within +/-0.0052% when evaluating the first exponential integral, E1. To achieve this result, a methodology that makes use of analytically known limiting behaviors at either extreme of the domain is employed. Because the fit accurately captures limiting behaviors of the E1 function, more accuracy is retained when the fit is used as part of the scheme to evaluate higher-order exponential integrals, E(n), as compared with the use of brute-force fits to E1, which fail to accurately model limiting behaviors. Furthermore, because the fit is compact, no special accommodations are required (as in the case of spliced piecewise fits) to smooth the value, slope, and higher derivatives in the transition region between two piecewise domains. The general methodology employed to develop this fit is outlined, since it may be used for other problems as well.