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 !
2026-03-24 20:38:03.1774384683
Rouche's Theorem of functions other than polynomial
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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.