Let the $\Omega$ constant be defined by the following integral:
$$\frac1{1+\Omega}=\int_\mathbb R\frac{\mathrm dt}{(e^t-t)^2+\pi^2}$$
How does one go about showing that $\Omega$ is the real solution to
$$1=\Omega e^\Omega$$
I tried starting by considering a semicircle contour and taking residues,
$$e^z-z=\pi i\\\implies z=-W_k(1)-\pi i$$
This has imaginary part greater than $0$ when $k\ge1$ where $W_k(z)$ is the Lambert W function, hence,
$$\int_\mathbb R\frac{\mathrm dt}{(e^t-t)^2+\pi^2}=2\pi i\sum_{k\ge1}\operatorname{Res}\limits\limits_{z=-W_k(1)-\pi i}\left(\frac1{(e^z-z)^2+\pi^2}\right)$$
But I haven't the slightest clue how I would go from here.
Any simpler ideas?
