The numerical computation of two transcendental functions related to the exponential integral

Abstract:

Algorithms for the computation of numerical values of the two transcendental functions \[ \int _0^x {\tfrac {1}{t}} [\operatorname {Ei} (t) - \gamma - \ln \left | t \right |]dt\quad {\text {and}}\quad \int _0^x {\tfrac {1}{t}} [\operatorname {Ei} (t) - \gamma - \ln \left | t \right |]\exp ( - t)dt,\] where $\gamma$ is Euler’s constant and $\operatorname {Ei} (t)$ is the exponential integral, are presented for all ranges of the real variable $x$. A table of values of these functions is also given.