Understanding the Algebra of Branch Cuts in Multi-Valued Functions

[sponsoredwalk]

How does one deal with finding the branches & branch cuts of a multi-valued function of a complex variable that is itself 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, i.e. the branch cut of a sum is the sum of the branch cuts or whatever the rule is, in general? It really would be fantastic if someone could help me out with this, I have no idea myself & can't find it discussed anywhere.

 

[fzero]

Say that the function $f(z)$ is defined on the domain $X$. We choose the branch cuts $B$ in such a way that $f(z)$ is single-valued on $X-B$. If

$$f(z) = \sum_i f_i(z),$$

suppose that each $f_i(z)$ has a branch cut $B_i$. For completeness of the discussion, we can allow that a $B_i$ could be empty if a $f_i$ is single-valued everywhere on $X$. Then $f(z)$ should be single-valued on $X-B$, where

$$ B = \cup_i B_i.$$

In other words, we take the set of individual branch cuts and find the subset of $X$ that includes all of them. This is, as you say, roughly the sum of the branch cuts. The important thing is that the total function is single-valued on the resulting domain after removing the appropriate branch cuts.

 

[jackmell]

How does one deal with finding the branches & branch cuts of a multi-valued function of a complex variable that is itself 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, i.e. the branch cut of a sum is the sum of the branch cuts or whatever the rule is, in general? It really would be fantastic if someone could help me out with this, I have no idea myself & can't find it discussed anywhere.

There is in my opinion only one way to get a good handle on this subject: draw them. Take for example \log(z-1)+\sqrt{z}. What does it look like? Not easy but one you have in hand a real-time interactive graphics utility to draw and explore them, then you can begin to really understand branch cuts, branch points, and the geometry of multivalued functions.
That plot below is a section of the imaginary component of the function. Yeah, it looks like a mess because it's static. But if you could rotate it, enlarge it, disect it piece by piece, draw contours over it, understand how to integrate over it, and explore it carefully and analyze it and other multivalued functions, you would slowly cultivate a deep understanding of this difficult subject.

Yeah, I have a tool for that, obviously. But it's messy code, needs tweeking for some functions like this one to get a good picture, and requires Mathematica.

 

[Bacle2]

To add to the previous, you may want to fix a branch of logz before figuring out the branches. Then it becomes a matter of using composition of functions : given fog , you want the image of g to not land on the branch cut chosen for f.

More complicated is the product.