The exponential integral $\mathrm{ei}(\lambda x)$ and its associated functions $\mathrm{ei}_{+}(\lambda x)$ and $\mathrm{ei}_{-}(\lambda x)$ are defined as locally summable functions on the real line and their derivatives are found as distributions. The convolutions $x^{r}\mathrm{ei}_{+}(x) \ast x^{s}e_{+}^{x}$ and $x^{r}\mathrm{ei}_{+}(x) \ast x^{s}e^{x}$ are evaluated.