I need to prove that $e^z + z$ is a bijection in the left half side of the complex plane.
I thought about showing that the line integral of this function over any curve that starts in $z$ and ends in $w$ is 0 iff $z=w$, thus proving that $e^z + z^2 / 2 =e^w + w^2 / 2$ iff $z=w$.
But this is all I have. I'd be happy with a hint.