Rouche's Theorem of functions other than polynomial

Question

All problems of Rouche's theorem are about zeros of polynomials ,

Is there another functions can I apply Rouche's Thm. to find the number of zeros other than polynomials ?

please give me an example because google is useless !

Answer 1

Sure. Suppose, for intance, that you wish to prove that the function $f(z)=e^z+5z$ has one and only one zero when $|z|<1$. Apply Rouché's theorem: if $|z|=1$, then$$|e^z|=e^{\operatorname{Re} z}\leqslant e<5=|5z|.$$Therefore, $f(z)$ has as many zeros in the region $|z|<1$ as the function $z\mapsto5z$. That is, it has one and only one.