Using Rouche's theorem prove that $\tan z = az$ has:
- infinitely many roots;
- only two purely imaginary roots if $0< a< 1$;
- all real roots if $a \ge 1$.
What do I take $f(x)$ and what do I take $g(x)$ (in general) for showing infinitely many roots and purely imaginary roots? I basically know showing finite roots within some region $|z| = a$
Also I think we can deduce with $\displaystyle \arctan z = \frac{1}{2i} \log \left( \frac{1 - iz}{1 + iz}\right) $ but how can it get it from Rouche's theorem?
