Using Rouche’s Theorem find the number of points $e^{z}+3z=0$.

Question

Use Rouche’s Theorem to determine for how many points $z\in D(0,1)$ we have $e^{z}+3z=0$.

Answer 1

Take $f(z)=3z$ and $g(z)=e^z$. Then $|g(z)|<|f(z)|$ for all $z$ in $C: |z|=1$. $$|g(z)|=|e^z|=e^{Re\; z}\leq e<3=|f(z)|$$ So Rouche's theorem gives that $f$ and $f+g$ must have same number of zeros inside of $C$. Clearly, $f$ has only one zero inside of $C$.