How to find value of each branch

Question

There is this question about the multi value operation of $(z-i)^{3/4}$ where it ask me to find the value of each branch when $z=1$. How do i approach it?

Thanks all..

Answer 1

First you have to find the absolute value and argument of your $u=z-i$ number (for $z=1$ it will be $|u|=\sqrt2$ and $\phi=-\frac{\pi}{4}$)

Now you just have to use the formula

$$\left(u^{\frac{a}{b}}\right)_k = |u|^{\frac{a}{b}}\left(\cos \frac{a\phi + 2k\pi}{b}+i\sin\frac{a\phi + 2k\pi}{b}\right)$$ for $k=0,1,...,b-1$