I have a problem understanding the following procedure. ( It's from a script)
Consider the domain C[0,$\infty$) and the branch of the logarithm given by
$ log(z)=ln(|z|)+i \cdot arg(z)$ ,with $arg(z) \in(0,2\pi)$.
Then we find: $z^{\frac{1}{2}} = \sqrt{|z|} e^{i arg(z)/2}$, where $arg(z^{\frac{1}{2}}) \in (0,\pi)$ The square root of any complex number that is not a positive real then lies in the upper halfplane and in this case we find $(-1)^{\frac{1}{2}}=i$
I don't understand why the last one follows. Why applies $(-1)^{\frac{1}{2}}=i$ ?
I know I can write: $-1^{\frac{1}{2}}= e^{\frac{1}{2}ln(|-1|)+arg(-1)}=e^{\frac{1}{2}arg(-1)}$ and arg(-1) ist $\pi$ and with that the upper equation would be correct....BUT that's actually wrong, since $\pi$ is not in the domain of arg. Can someone explain that to me?