MAT334--2020F > Test 1

2020 TT1 Main Setting - Problem 1b

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A A:
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

RunboZhang:
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 non-negative $n$, our $z$ is in the first quadrant of complex plane.

A A:
Thank you!!