I'm looking for the solution to the equation $E_1(x)=e^{-x}$, where $E_1(x)$ is the exponential integral function. The solution is approximately 0.434818204, but is this constant well known/studied?
The reason I think this constant should be important is that exponential integral, which is upper bounded by $e^{-x}/x$, behaves like $e^{-x}$ for larger $x$ and like $-\log(x)$ for smaller $x$, so roughly this point (and, for that matter, when $E_1(x)=-\log(x)$, around $x \approx 0.676355077$) are natural "transition points" to study. Furthermore, the function $x E_1(x)$ is a "sharpened" version of $x e^{-x}$ and $x \log(x)$ and its maximum is attained at the point I'm interested in (whereas maximums of the other two functions are trivially attained at $x=1$ and $x=e$).
In general the elementary functions $x e^{-x}$ and $x \log(x)$ are well understood (e.g., Lambert W function tries to capture many complexities of the former), whereas there seems to be not much known about $x E_1(x)$ which exhibits a mixed behavior.
