Toronto Math Forum
MAT3342020F => MAT334Tests and Quizzes => Test 1 => Topic started by: A A on October 14, 2020, 05:42:27 PM

Hello, can anyone explain the solution to Problem 1b: which of the complex roots are in the first complex quadrant? Why is it all zn that are in the first complex quadrant?
Note (for context): Problem 1a: Find all the complex roots of the equation tanh(3z) = 1 + 2i

Firstly, for the first quadrant, we have $Re(z) > 0$ and $Im(z) > 0$.
Secondly, we have $z = \frac{1}{12}log(2) + (\frac{\pi}{8} + \frac{2\pi }{6} n)i$, $n \in \mathbb{Z}$ by part (a).
By combining the previous two conclusions we have, $Re(z) = \frac{1}{12}log(2) > 0$ since $log(2) > 0$. Also $Im(z) = (\frac{\pi}{8} + \frac{2\pi }{6} n) > 0$ when $n \ge 0$.
Therefore, as long as we have a nonnegative $n$, our $z$ is in the first quadrant of complex plane.

Thank you!!