I would like to solve the complex equation $(z-1)^3=9(\overline{z}-1)$.
Here's what I did:
- Attempt 1: $w=z-1$, $w^3=9[cos(-\alpha)+isin(-\alpha)]-9$
- Attempt 2: $(x+iy-1)^3=9(x-iy-1)$
- Attempt 3: $z^3-3z^2+3z-1=9(\overline{z}-1)$
But it seems like neither of those attempts takes me to the solutions.
With $y=z-1$ you have $$y^3=9\bar y$$
Thus $$|y^3|=9|y|$$ We get $|y|=0$ or $|y|=3$
Thus $y=0$ or $y=3e^{i\theta }$
Plugging in $$y^3=9\bar y$$ gives us $e^{4i\theta}=1$
You can continue from there.