I have recently encountered a differential equation whose solution has a term \begin{equation} \frac{1}{2}e^{-\frac{1}{2 \varepsilon} e^{i \tau}} \int_{\tau_0}^\tau e^{\frac{1}{2 \varepsilon} e^{i \tau'}} d\tau' \text{ where } \varepsilon \ll 1. \end{equation} I believe this integral can be evaluated in terms of the exponential integral, either $\operatorname{Ei}$ or $\operatorname{E}_1$. However, since this term is part of a solution to a differential equation it must depend smoothly on $\tau$, and the exponential integral I know to have branch cuts. I therefore ask: what is the Riemann surface of the exponential integral?
2026-03-27 04:57:09.1774587429
What is the Riemann surface of the exponential integral?
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