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.

 

[blue_raver22]

I understand the importance of understanding the algebra of branch cuts in multi-valued functions. This concept is crucial in order to accurately analyze and interpret the behavior of these functions. In the case of finding the branches and branch cuts of a multi-valued function of a complex variable that is the sum of two multi-valued functions, there are several key steps one can take to understand and deal with this problem.

First, it is important to identify the branch points of each individual function. These are the points where the function becomes multi-valued, and they are typically located at the roots of the function's polynomial or at points where the function is undefined. In the examples given, the branch points for each function would be at z=0 and z=1 for the square root functions, and at z=1 for the logarithm function.

Next, we can use the rules for combining branch cuts to determine the overall branch cut for the sum of two functions. In general, the branch cut for the sum of two functions is the union of their individual branch cuts. This means that for the first two examples, the branch cut would be the line connecting z=0 and z=1, while for the third example it would be the union of the lines connecting z=0 and z=1, and z=0 and z=1-i. For the fourth example, the branch cut would be the line connecting z=1 and infinity.

It is also important to note that the branch cuts for each individual function may need to be adjusted in order to properly combine them. This can be done by introducing branch points at the intersection of the individual branch cuts. For example, in the first two examples, the branch cut for the square root functions would need to be adjusted at z=1 in order to properly combine them.

In summary, understanding the algebra of branch cuts in multi-valued functions requires identifying the branch points of each individual function and using the rules for combining branch cuts to determine the overall branch cut. Adjustments may need to be made in order to properly combine the individual branch cuts. I hope this explanation helps in your understanding of this concept.