Inverse function of $\operatorname{li}(x)$ over $x>\mu$?

Solution 1:

There are two answers. There is such an inverse function, and it is real analytic. However, as J. M. indicates, there is no evidence, in the usual places, that anyone has found an attractive asymptotic expansion for the inverse of the exponential integral function. There is a very careful treatment of this in PECINA

Just to include one item I like, for $x > 1,$ from 5.1.10 in Abramowitz and Stegun, we have $$ \mbox{li} \; x = \gamma + \log \log x + \sum_{n=1}^\infty \; \frac{(\log x)^n}{n \, n!} $$ where $\gamma = 0.5772156649...$ is the Euler-Mascheroni constant.