For a fixed complex number $\displaystyle \lambda$, show that if $\displaystyle m$ and $\displaystyle n$ are large integers, then the equation $\displaystyle e^z=z+\lambda$ has exactly $\displaystyle m+n$ solutions in the horizontal strip $\displaystyle \{ -2\pi i m < \text{Im}(z)<2 \pi i n \}$.

The hint says to consider $\displaystyle f(z)=e^z -z-\lambda$ and sketch the image of the boundary of $\displaystyle \{ -R< x<R, -2 \pi m < y < 2 \pi n \}$. Keep in mind $\displaystyle |\lambda | << m, n << R$.

I do not see how to prove this now. I need some help on this one. Thank you.