Exponential integral

Summary

The exponential integrals are the functions $E_1(x) = \int_x^\infty \frac{e^{-t}}{t}\, dt$ for $x > 0$ and $\operatorname{Ei}(x)$, the principal value of $\int_{-\infty}^x \frac{e^t}{t}\, dt$ for $x \ne 0$. They are related by $E_1(x) = -\operatorname{Ei}(-x)$ for $x > 0$, and $E_1(x)$ is the upper incomplete gamma function $\Gamma(0, x)$. The logarithmic integral is $\operatorname{li}(x) = \operatorname{Ei}(\ln x)$. They occur in heat transfer, in groundwater flow and in radiative transfer.