Let $$e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!} $$ denote the exponential function, which is defined on the entire complex plane.
There is a fixed point of this function at $w= a+bi$ where $a \approx 0.31813$ and $b \approx 1.33723$.
I am guessing there is no closed form expression for $w$. If there is, please correct me. But if there is not a closed form expression, how does one go about proving something like that? Any pointers to the appropriate literature would be appreciated.
$e^z=z$, so $ze^{-z}=1$ or $-ze^{-z}=-1$. Let $u=-z$, so the equation becomes $ue^{u}=-1$. This equation can be solved via the Lambert-W function, getting $u=W(-1)=-0.31813\ldots-1.3372\ldots i$, which upon negation gives you your answer. There are some power series expansions of $W$ but no easy closed form for the solution.