I can't interpret this branch cut choice....

• Jan 3rd 2014, 03:53 AM
Mengqi
I can't interpret this branch cut choice....
Hi,

I'm reading a PhD dissertation. The author mentioned a branch cut when calculating the multivalued function $k^2$, where k is a complex number, and he introduced a small number $\epsilon$ to form $\sqrt{k^2+\epsilon^2}=\sqrt{k+i\epsilon}\sqrt{k-i\epsilon}$. 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 $\epsilon$ approaches zero, $\sqrt{k^2}=k$ when $Re(k)>0$ and $\sqrt{k^2}=-k$ when $Re(k)<0$.

My question is how does the author get this conclusion? i.e., $\sqrt{k^2}=k$ when $Re(k)>0$ and $\sqrt{k^2}=-k$ when $Re(k)<0$ with this branch cut. For another example, the branch cut of $\sqrt{z^2-1}$ 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.
• Jan 4th 2014, 01:33 PM
Mengqi
Re: I can't interpret this branch cut choice....
Anybody has any clues? Or my question is not clear? Thanks.
• Jan 5th 2014, 06:12 AM
Mengqi
Re: I can't interpret this branch cut choice....
Never mind. I have figured it out.