In Buhler et al. (Math Comput 44(170):473–481, 1985), there are given two representations of the function \(G_{1}(x)\) (equal to the exponential integral \(E_{1}(x)\) ) that appear in an expression of the first derivative of the L-function of an elliptic curve defined over \(\mathbb {Q}\) at 1. One is the Puiseux Series, and the other is the continued fraction representation. In Hitotsumatsu (Introduction to Special Functions, Morikita Publishing Co. Ltd., Chiyoda, 1999), we can see how to construct formally this continued fraction from \(F(x):=e^{x}E_{1}(x)\) (explained briefly in Introduction), but we have never seen a proof of that it converges to the original function F(x). More precisely an asymptotic expansion of \(F(x)=e^{x}E_{1}(x)\) is \(\displaystyle \sum _{k=1}^{\infty } (-1)^{k-1}(k-1)!\left( \displaystyle \frac{1}{x}\right) ^{k}\) , and this gives the continued fraction by quotient-difference algorithm, which is briefly announced by E. Stiefel and developed by Rutishauser (Zeitschrift für Angewandte Mathematik und Physik 5:233–251, 1954). In this paper we define “a continued fraction expansion of F(x) at infinity”, which is analogous to the regular continued fraction expansion of real numbers, and prove that this expansion gives the same continued fraction. Moreover, we give concrete representations of rational functions which are obtained by truncating the continued fraction.
