The Exponential and Logarithmic Integral

This entry provides three closely related special functions: the exponential integral $\text{Ei}$, the logarithmic integral $\text{li}$, and the complementary exponential integral $\text{Ein}$. All three functions are defined both on the reals and in the complex plane.

Basic properties are shown, e.g.:

• derivatives, continuity, limits, etc.
• $\text{Ei}(x) = \int_{-\infty}^x e^t/t\,\text{d}t$ and $\text{li}(x) = \int_0^x 1/\ln t\,\text{d}t$ for real $x$, where for $x>0$ and $x>1$, respectively, the integrals must be interpreted as Cauchy principal values
• asymptotic expansions for $\text{Ei}$ and $\text{li}$
• the relationship to the incomplete gamma function via $\Gamma(0, x) = -\text{Ei}(-x)$ and $\Gamma(0, z) = \text{Ein}(z) - \ln z - \gamma$