zeroes of analytic function in a square below is a question I came across while studying for an exam, and I believe the way to handle it is to use the argument principle to count the increase in the argument, however, when I was reading an example of this in Gamelin's book, he only did an exampl which involved looking at arcs of circles and polynomials, which make this seems clear, but I've been unable to replicate his observations in my own setting. Would anyone mind giving me an outline of how you think one may approach this?
Question:  How many zeroes does the function $f(z) = e^z + z$ have in the square $[-10,10] \times [-6 \pi i, 6 \pi i]$?
 A: $f(z)$ is an entire function without double zeroes, since $f'(z)=e^z+1$.
The number of zeroes in the rectangle $R=[-10,10]\times [-6\pi i,6\pi i]$ is given by
$$ N(R) = \frac{1}{2\pi i} \oint_{\partial R} \frac{f'(z)}{f(z)}\,dz = \frac{1}{2\pi i}\oint_{\partial R}\frac{1-z}{e^z+z}\,dz\tag{1}$$
where:
$$ \int_{-10-6\pi i}^{10-6\pi i}\frac{1-z}{e^z+z}\,dz=\int_{-10}^{10}\frac{1-z+6\pi i}{e^z+z-6\pi i}\,dz\tag{B} $$
$$ \int_{10+6\pi i}^{-10+6\pi i}\frac{1-z}{e^z+z}\,dz=\int_{-10}^{10}\frac{-1+z+6\pi i}{e^z+z+6\pi i}\,dz\tag{T} $$ 
$$ \int_{10-6\pi i}^{10+6\pi i}\frac{1-z}{e^z+z}\,dz = \int_{-6\pi i}^{6\pi i}\frac{-9-z}{e^{z+10}+z+10}\,dz\tag{R} $$
$$ \int_{-10+6\pi i}^{-10-6\pi i}\frac{1-z}{e^{z}+z}\,dz = \int_{-6\pi i}^{6\pi i}\frac{z-11}{e^{z-10}+z-10}\,dz\tag{L} $$
where by numerical approximations we get that the contribute provided by the bottom and top side of $\partial R$ is $(B)+(T)\approx 4i$ and the contribute provided by the left and right side of $\partial R$ is $(L)+(R)\approx 40i$. It follows that $f$ has $\color{red}{7}$ zeroes inside $R$. These roots are given by the solutions of
$$\left\{\begin{array}{rcl}e^a \cos(b)+a &=& 0 \\ e^a\sin(b)+b &=& 0\end{array}\right. $$
with $a\in[-10,10]$ and $b\in[-6\pi,6\pi]$. From $\cos(b)=-ae^{-a}$ we get $a\geq -1$, so the original rectangle $R$ can be substantially resized, too. The problem boils down to counting the intersections between the blue curve and the purple curve in the range $(a,b)\in[-1,10]\times[-6\pi,6\pi]$:
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You can check I was not lying, they really are seven.
