Branch points and cuts
It’s unbelievable how hard it is to find a good working definition of a branch point– I have looked through more than 15 books of varying sophistication, and am still at a loss.
The simpler complex analysis books substitute handwaving for a rigorous definition, and implicitly give the message that you should be able to look at an expression for a function and quickly find the branch points. But notice how all the examples are contrived: 
or 
. And notice also the way they tip-toe around the issue of branch points at infinity: the standard approach here seems to be to define infinity to be a branch point of a function 
if 
is a branch point of 
. Besides begging the question of what makes 0 a branch point, why is this a reasonable/useful definition? Why do I care about branch points at infinity? I read in one book that the identifying character of a branch point at infinity is, if you travel from a point back to itself in a really big circle around the origin, you get a different value starting than the one ending. What the hell? Isn’t that the ‘definition’ given for a branch point at 0?
The more advanced complex analysis books invariably give a reasonable sounding definition, closer to the end of the book than the beginning or even the middle. Said definition relies on some heavy concepts like local biholomorphicity or holomorphic mappings between Riemann surfaces, or analytic continuation. Hence they are practically useless to me.
What’s the problem here? Is there no simple exposition of the basics of branch points: what they are, how to spot them, how to make cuts, etc.?
The closest I’ve found to a reasonable explanation of branch points was given in Visual Complex Analysis, where the author gives a neat diagram illustrating why (sort of) 0 is a branch point of 
:
In this image, 
is the starting point, 
are the different values of 
, and 
are closed paths that are traveled from back to . His point here is that traveling these paths in the 
domain means the image in the 
domain travels a corresponding path at a radius 
as large, and with 
the angular speed. Notice how paths 
start off at 
in the image domain, but after transversal wind up at different values of , corresponding to the number of circuits they take around 0.
I don’t see why 0 has this special property, or how to identify it as a potential branch point from just looking at the formula , but at least this author isn’t expecting me to mine through crap looking for gold.
Possibly relevant posts:
- Where’s the error? (8/14/2007)
- Complex Analysis pt. I (11/3/2003)
- Bombelli father of complex arithmetic? (5/26/2005)
I’m not overly clear on this either, but my current thinking is that the branch point is the end point of the branch cut, and so if you’ve wound around it, you’ve gone from one copy of the C plane to another. Once you’re on the second copy of C, the answers you get will be the second set of answers to the problem, so when you return to you “original” point, you’re really on another sheet, so the answer will be different. But there are multiple choice for a branch cut! But, I think they should all have the same branch point. Hence winding around the branch point in the domain plane will take you to a different point in the range plane.
The above probably made no sense, and the following links will likely be of little help, but I’ll post them anyway. These pages can at least be said to have pretty pictures, though exactly what this guys is doing escapes me. He’s big on rhetoric, short on analysis, but the animations are good.
Comment by ObsessiveMathsFreak — 10/10/2006 @ 3:30 am
A simple way that I keep it in mind (may not be what you are looking for), is that a branch point is “cut” in the graph. If we do not make the cut, the graph will repeat itself forever. For example, if I had a circle in a plane, represented by CosX. If I limit X to 0-2pi, that corresponds to one rotation. If I don’t limit x, the circle keeps going around and around and around forever. 0-2pi is then my branch point, and outside of this domain, the function repeats and is not analytic.
Comment by Adam Hughes — 11/19/2006 @ 3:04 pm
I like the second explaination. and would like to elaborate more with a way to visualize it. if you walk along aa curve through a point, you have a certain gradient (slope) G1, and if you walk in the opposite direction you also have another G2. if G1 and G2 are crossing each other, then you have an abrupt change, then you have a discontinuity, then you have branch point. Also if you can determine one G and can’t determine the other because it’s at infininty, then it’s also a branch point. PS: I advise you to verify this idea, but it sounds like a good generalization.
Comment by John Bebawy — 10/13/2007 @ 7:10 pm
To see why there’s a branch point at infinity, I think it’s most helpful to think about the projective plane, since you can “see” the complex infinity better there. Since there is only one infinity on the complex plane, you can think of it as 0 being at the south pole and infinity at the north pole. So we then can easily see that a branch point at 0 often will imply a branch point at infinity since the reason for the break is often that you get multiple values as you rotate around 0 (just expand the circle until it’s around infinity instead of around 0).
Comment by Gordon Worley — 11/23/2007 @ 8:02 am