# Algebra of Branch Cuts

How does one deal with finding the branches of a multi-valued function of a complex variable that is the sum of two multi-valued functions, something like the following:

$f(z) = \sqrt{z} + \sqrt{1 - z}$,

$f(z) = \sqrt{z} + \sqrt{z - 1}$,

$f(z) = \sqrt{z} + \sqrt{z(z - 1)}$

$f(z) = \log(z - 1) + \sqrt{z}$?

In other words: how does one understand the algebra of branch cuts and branch points in general?

• Do you mean understanding where the branch cuts are, or an algebra of branch cuts? Apr 20, 2013 at 3:52
• Apologies, I meant finding them. Apr 20, 2013 at 3:53
• possible duplicate of How does a branch cut define a branch? Apr 20, 2013 at 4:01
• There's nothing on the algebra of branch cuts in any other thread on here, the closest is this question but they never addressed the sum of branch cuts, let alone the algebra (arithmetic?) of this in general, could you remove the possible duplicate stuff? Apr 20, 2013 at 4:31
• @speedwalk Your post doesn't incluse a question about the sum of branch cuts either. You specifically explained to me that you didn't mean the algebraic structure of the branch cuts. If that is what you want to ask about, that's fine, but you're going to need to edit the wording to make this clear. Apr 20, 2013 at 6:07