Finding complex zeros of an holomorphic function I was studying a simple function when I had problem to determine the number "$n$" of zeros in the set $\{z \in \mathbb{C}: Re(z)<0   \}$ of the function $f: \mathbb{C} \to \mathbb{C}$ defined:
$$f(z)=e^z+z^5+3$$
where $Re(z)$ means the real part of $z$.
I tried to use the Residue Theorem computing $\int_{ \partial R} \frac{f'(z)}{f(z)}dz=n$ where $R \subset \{z \in \mathbb{C}: Re(z)<0   \}$ is a compact set sufficiently big but I had problem to solve it.
Any help will be appreciate.
 A: Let $f(z)=e^z+z^5+3$
We can do this in two steps as we notice that for $|z| \ge 2, \Re z <0, |e^z+z^5+3| > 28$ since $|e^z| <1$ so all the roots in the left half-plane are inside the circle $|z| \le 2$
Applying Rouche on the disc $\bar D(0,2)=|z| \le 2$ where $|e^z+3| \le e^2+3<32=|z|^5$ it follows that $f$ has $5$ roots inside $D(0,2)$ and we need to decide how many are in the half-disc with $\Re z <0$ (obviously there will be $1,3,5$ as $f$ has a unique (negative) real root and the complex one are conjugates)
But now we consider the domain $U$ bounded by the part of the circle $|z|=1, \Re z \le 0$ the segments from $i, 2i$ and $-i, -2i$ on the imaginary axis and the part of the circle $|z| \le 2, \Re z \ge 0$. Since $|e^z+z^5+3| >3-1-1>0$ for $|z| \le 1, \Re z \le 0$ the roots of $f$ in the left half-plane are in $D(0,2)- U$, so they are equal to $5$ minus the number of roots of $f$ in $U$
But now we claim that on $\partial U$, $z^5+3 \ne 0, f+\lambda (z^5+3) \ne 0, \lambda \ge 0$ which implies by the symmetric form of Rouche ($(1-t)f+t(z^5+3), 0 \le t \le 1$ is a homotopy from $f$ to $z^5+3$ that is never zero on $\partial U$ by the conditions above) that $f$ and $z^5+3$ have the same number of roots inside $U$ and hence there are $2$ such by looking at the roots of $z^5+3$, so the final answer is that $f$ has $3$ roots in the left half-plane
So let's prove the claim above - first $|z^5+3| >2$ on the left part of $U$, it is clearly not zero on the imaginary axis and it is at least $29$ on the right part of $U$
Similarly since $g_{\lambda}(z)=f(z)+\lambda (z^5+3)=e^z+(\lambda +1)(z^5+3), \lambda \ge 0$ we have that $|g_{\lambda}| \ge 2(1+\lambda)-1 >0$ on the left part of $U$ and it is at least $29(1+\lambda)-e^2>0$ on the right part of $U$, while on the imaginary axis its real part is at least $3(1+\lambda)-1>0$ since $z^5$ is purely imaginary there and $|\Re e^z| \le |e^z|=1$ so we are done!
