Use Rouche's theorem to show that $e^z$ never vanishes on the unit disk Use Rouche's theorem to show that $e^z$ cannot vanish on the unit disk.
Generally, in applications of Rouche's theorem, if I was trying to prove $f$ had a certain number of zeros in the unit disk, I would find holomorphic functions $g$ and $h$ such that $g + h = f$, show that $|g| > |h|$ on the unit circle, and then conclude that $f$ had the same number of 0's as $g$. However, I 'm not sure what functions $g$ and $h$ I would use, in this case... I know that $e^z = e^x \cos y + ie^x \sin y$, but I can't show that either one of those terms is bounded by the other on the unit circle. Am I missing something, or should I be applying Rouche's theorem in a different way?

Take $f(z)=e^z$ and $g(z)=k$ with $k\in \Bbb{R},\,k\gt {e^2\over 2}$ we have $|e^z-k|\lt k$ on the unit circle and the exponential has the same number of zeros as the positive constant on the unit disk
