circlescomplex numbersinverse
How is moving $\frac{1}{z}$ in complex plane if $z$ is described by a circle which has radius $r$ and center $a+b*i$
I've just started complex algebra and still having some trouble imagining it.
How one does even solve this kind of problems, I don't want the complete solution I just want some appropriate ways for working with complex numbers to solve this kind of problems.
Thank you.)
Because $|\frac{2}{w}-(1+i)|=2$, so $\bigg(\frac{2}{w}-(1+i)\bigg)\bigg(\frac{2}{\bar{w}}-(1-i)\bigg)=4$
$$\therefore \frac{4}{w\bar{w}}-(1-i)\frac{2}{w}-(1+i)\frac{2}{\bar{w}}+2=4\Rightarrow w\bar{w}-(1-i)w+(1+i)w=2$$ $$\Longrightarrow|w+(1-i)|^2=4$$
So we get $\frac{2}{w}$ is actually a circle with center $-1+i$ and radius $2$.
$(z+1)^4-z^4=((z+1)^2+z^2)((z+1)^2-z^2)=(2z^2+2z+1)(2z+1)$, which gives the answer: $$\left\{-\frac{1}{2}-\frac{1}{2}i,-\frac{1}{2}+\frac{1}{2}i,-\frac{1}{2}\right\}$$
Best Answer
Hint: Compute $\dfrac1{r\bigl(\cos(\theta)+i\sin(\theta)\bigr)}$.