Evaluate $-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$ in $\gamma$.

Question

Evaluate $\gamma$ expressed, involving Lambert function, by $$-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$$ where $\gamma<1$. I doubt that it is possible to find a value for $\gamma$ in closed form. Numerical solutions or estimates are welcome.

Answer 1

Exact expression for the solution is: $$\gamma=\frac{\pi/4}{e^{\pi/4}-1}=0.6581842...$$

EDIT The two branches are related in a trascendental way. Their difference can be used to solve other trascendental equations.

If $y(x)=\frac{x}{1-e^x}$

then :

$x=W_0(ye^y)-W_{-1}(ye^y)=y-W_{-1}(ye^y)$ for $-1< x <0 $

$x=W_{-1}(ye^y)-W_{0}(ye^y)=y-W_{0}(ye^y)$ for $x < -1 $

See

http://www.apmaths.uwo.ca/~djeffrey/Offprints/SYNASC2014.pdf

https://mathoverflow.net/questions/195923/equation-between-the-two-branches-of-the-lambert-w-function

Answer 2

$$\gamma\simeq0.658184273600902103771558680728257154392531172950\ldots$$