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

How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$?

Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and $\operatorname{li}(\mu)=0$.

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$\mu$ here is the Ramanujan-Soldner constant. – J. M. Dec 29 '11 at 3:51
$\mathrm{li}(z)=\mathrm{Ei}(\ln\,z)$; your problem here is computing the inverse of the exponential integral. – J. M. Dec 29 '11 at 3:52
Thanks but I don't understand clearly. How can I compute the inverse of the exponential integral? is it some numerical way? – chimpanzee Dec 29 '11 at 4:18
That's the problem. I don't see an easy way to derive a nice approximation for the exponential integral's inverse. – J. M. Dec 29 '11 at 4:25

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.