Abstract
Exponential integral for real arguments is evaluated by employing a fast-converging power series originally developed for the resolution of Grandi’s paradox. Laguerre’s historic solution is first recapitulated and then the new solution method is described in detail. Numerical results obtained from the present series solution are compared with the tabulated values correct to nine decimal places. Finally, comments are made for the further use of the present approach for integrals involving definite functions in denominator.
Highlights
Exponential integral is encountered in various physics problems such as heat transfer, scattering, and neutron transport processes
Numerical treatment of a generalized exponential integral based on rational minimax approximations and a comprehensive account of the relevant literature can be found in Milgram [4]
For x = 0.25 the maximum possible n is 10 and the corresponding relative error percentage is 1.56%1. This point is the only drawback of the present approach
Summary
Exponential integral is encountered in various physics problems such as heat transfer, scattering, and neutron transport processes. Standard solutions to the integral are typically given in series forms with restrictions on the magnitudes of arguments or asymptotic expansions [1]. Laguerre (1834-1886) gave a continued fraction expansion for its solution in [2], which was reprinted in [3]. Numerical treatment of a generalized exponential integral based on rational minimax approximations and a comprehensive account of the relevant literature can be found in Milgram [4]. Based on fast-converging series expansions the present work constructs a solution uniformly valid for the entire domain of positive real arguments
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