The efficient computation of some generalised exponential integrals

AbstractThe accurate and efficient computation of the special functions Gk(x) is discussed, whereGk(x)=1(k−1)!∫1∞exp(−xy)(logy)k−1dyy.These functions appear in the computation of the derivatives of the L-series of an elliptic curve, and in radiative transfer problems from astrophysics. By dividing (0,∞) into 3 sub-intervals, we derive Chebyshev polynomial expansions for Gk,k=1,…,4 with the coefficients given to an accuracy of 20 decimal places