I'm trying to find the number of zeros for the function $f(z) = z + 2 - e^z$ in the half plane $\{\mathscr{R}z < 0\}$.
I know I'm supposed to use Rouche's theorem, which states that if both $f$ and $g$ are holomorphic inside and on some closed contour $C$, and $|f(z)| > |g(z)|$ for all $z \in C$, then $f$ and $f+g$ have the same number of zeros inside of $C$.
The problem is, I'm having some trouble coming up with intuition about what to choose for $f$ and $g$. Something tells me that $f(z) = z+2$ and $g(z) = -e^z$, but how precisely do I complete the problem with Rouche's theorem?
Thanks.