Hi,

I'm reading a PhD dissertation. The author mentioned a branch cut when calculating the multivalued function $\displaystyle k^2$, where k is a complex number, and he introduced a small number $\displaystyle \epsilon$ to form $\displaystyle \sqrt{k^2+\epsilon^2}=\sqrt{k+i\epsilon}\sqrt{k-i\epsilon}$. The branch cut is chosen as shown in the figure attached. And the author then asserts that the branch cut so chosen is because he wants to have when $\displaystyle \epsilon$ approaches zero, $\displaystyle \sqrt{k^2}=k$ when $\displaystyle Re(k)>0$ and $\displaystyle \sqrt{k^2}=-k$ when $\displaystyle Re(k)<0$.

My question is how does the author get this conclusion? i.e., $\displaystyle \sqrt{k^2}=k$ when $\displaystyle Re(k)>0$ and $\displaystyle \sqrt{k^2}=-k$ when $\displaystyle Re(k)<0$ with this branch cut. For another example, the branch cut of $\displaystyle \sqrt{z^2-1}$ would be (-1,1) because in this interval the values are different across the branch cut, while in other intervals like (-infty,-1) and (1, infty) the function is single valued.

Thanks.