Suppose that a branch of the function $f(z) = (z −1)^{\frac{2}{3}}$ is defined by means of the branch cut in Fig. 2 and that it takes the value $1$ when $z = 0$. Determine the value of $f(z)$ and of its derivative $f′(z)$ at the point $z=i$.
I don't know how to start?
From the figure we take $$i-1=\sqrt{2}e^{-i\pi/4}(0-1)\ ,$$ hence $$f(i)=2^{1/3}e^{-i\pi/6}f(0)=2^{1/3}\left({\sqrt{3}\over2}-{i\over2}\right)\ .$$ Furthermore from $f^3(z)=(z-1)^2$ we obtain $$3f^2(z)f'(z)=2(z-1)$$ and therefore $$f'(i)={2(i-1)\over 3f^2(i)}=\ldots\quad.$$