Finding all roots of equations which solutions are different than $z=0$

Question

$$2\overline z = z^7 \, , \,\,32\overline z = z^7 \,,\,\, 128\overline z + z^7=0 $$

Let's say that $z=r\\$ then $2r=r^7$ then $2=r^6$ which gives us $r=\sqrt[6]{2}$. Now we take (argument) $$\text{arg}\, z=\alpha\,, \,\,\text{arg}\,\overline z=-\alpha\, , \,\,\text{arg}\,2\overline z=-\alpha$$ so $$-\alpha=7 \alpha \, , \,\,8\alpha=0\,,\,\,8\alpha=2k\pi\,,\,\, \alpha=k\frac {\pi}{4}.$$

Am I doing this the correct way? Can someone explain what am I supposed to do next? How can I solve other equations?

Answer 1

HINT

Multiplying by $z$ we obtain

$$ 2\overline zz = z^8 \iff z^8=2|z|^2 \implies |z|^6=2 \implies |z|=2^\frac16$$

and then

$$z^8=2^\frac43 \implies z=\ldots$$

Answer 2

Let me try to solve $$z^n = t\bar{z}^m$$ ($m$ and $n$ are positive integers). If $t=0$, the only solution is $z=0$. Suppose that $t\ne 0$. Clearly, $z=0$ is a still solution. We are seeking all solutions $z\neq 0$.

If $m=n$, we have $$\left(\frac{z}{\bar{z}}\right)^n=t.$$ Taking modulus we get $|t|=1$ is the only way to have a non-zero soln. If $t=e^{\theta i}$ for some $\theta\in [0,2\pi)$. That is, $$\frac{z}{\bar{z}}=e^{\theta i+\frac{2\pi k i}{n}}$$ for some $k=0,1,\dots,n-1$. That is, $$z=r e^{\frac{\theta i}{2}+\frac{\pi k i}{n}}$$ for some $k=0,1,\dots,n-1$ and $r\in\Bbb{R}$ s.t. $r\ne 0$.

If $m\neq n$, we have $$|z|^{n-m}=\left|\frac{z^n}{\bar{z}^m}\right|=|t|.$$ So, $|z|=|t|^{\frac{1}{n-m}}$. That is, $$z=|t|^{\frac{1}{n-m}}e^{\phi i}$$ for some $\phi\in[0,2\pi)$. Plugging this into $z^n=t\bar{z}^m$, we have $$|t|^{\frac{n}{n-m}}e^{n\phi i}=t|t|^{\frac{m}{n-m}}e^{-m\phi i}.$$ Let $t=|t|e^{\theta i}$ for some $\theta\in[0,2\pi)$. We get $$e^{(m+n)\phi i}=e^{\theta i}.$$ So, $\phi = \frac{\theta}{m+n}+\frac{2k\pi}{m+n}$ for $k=0,1,\dots,m+n-1$. I then get $$z=|t|^{\frac{1}{n-m}}e^{\frac{\theta i}{m+n}+\frac{2k\pi i}{m+n}}$$ for $k=0,1,\dots,m+n-1$.

Answer 3

All equations are of the form

$$a\overline z=z^7$$

with $a$ real.

Switching to polar coordinates,

$$|a|r=r^7$$ and $$\theta_a-\theta_r=7\theta_r+2k\pi$$ where $\theta_a$ is $0$ or $\pi$.

Then

$$z=\sqrt[6]{|a|}\text{ cis}\frac{\theta_a+2k\pi}8.$$