How to find all complex solutions?

Question

When i'm solving this Equation $$z^{2018} + |z|^{1995}* \bar{z}^{83} = 2$$

I got this: $$z_{k} = (cos(\frac{2\pi k}{1995}) + isin(\frac{2\pi k}{1995}) * \frac{1}{\sqrt[2018]{cos(\frac{4202\pi k}{1995})}}$$ $$k = 0 ... 1994$$ How to find all roots?

Answer 1

Let $z=re^{it}$

$$\implies2=r^{2018}(\cos2018t+i\sin2018t+\cos83t-i\sin83t)$$

Using Prosthaphaeresis Formulas, $$\dfrac2{r^{2018}}=2\cos\dfrac{(2018+83)t}2\left(\cos\dfrac{(2018-83)t}2+i\sin\dfrac{(2018-83)t}2\right)$$

$\implies\sin\dfrac{(2018-83)t}2=0\implies\dfrac{(2018-83)t}2=m\pi$ where $m$ is any integer

If $m$ is even, $=2n$(say), $\implies\cos\dfrac{(2018-83)t}2=+1$

$$t=\dfrac{2n\pi}{1935},0\le2n<1935$$

$$\dfrac2{r^{2018}}=2\cos\dfrac{2101}2\cdot\dfrac{2n\pi}{1935}\implies r=?$$

What if $m$ is odd, $=2n+1$(say) $\implies\cos\dfrac{(2018-83)t}2=-1$