1. ## Branch points

I am trying to find the branch points of w = Log(z(z-1)) and I know that Log is undefined for z = 0 and 1.

Let theta = arg(z) and phi = arg(z-1)

Log(z(z-1)) = ln[abs(z)abs(z-1)e^(itheta)e^(iphi)] + iArg[abs(z)abs(z-1)e^(itheta)e^(iphi)]

So, if I traverse a closed path abs(z) and abs(z-1) remain the same. And theta and phi are multiplied by an integer (1).

If I go around the points z = 0 and z = 1 (x = 1, y = 0), then obviously theta and phi both change from some angle to that angle + 2pi, so the value of w will be the same around closed path.

If I go in a loop between z = 0 and z = 1, starting at theta = 0, theta ranges from 0 to 2 pi and phi is 0 at start and finish. Once again, this just yields same value of w.

I can't think of any other paths, so I can't decide if there are or are not branch points because I can't find any paths for which the values of w disagree.

2. Originally Posted by amoeba
I am trying to find the branch points of w = Log(z(z-1)) and I know that Log is undefined for z = 0 and 1.

Let theta = arg(z) and phi = arg(z-1)

Log(z(z-1)) = ln[abs(z)abs(z-1)e^(itheta)e^(iphi)] + iArg[abs(z)abs(z-1)e^(itheta)e^(iphi)]

So, if I traverse a closed path abs(z) and abs(z-1) remain the same. And theta and phi are multiplied by an integer (1).

If I go around the points z = 0 and z = 1 (x = 1, y = 0), then obviously theta and phi both change from some angle to that angle + 2pi, so the value of w will be the same around closed path.

If I go in a loop between z = 0 and z = 1, starting at theta = 0, theta ranges from 0 to 2 pi and phi is 0 at start and finish. Once again, this just yields same value of w.

I can't think of any other paths, so I can't decide if there are or are not branch points because I can't find any paths for which the values of w disagree.
The branch points of $w = \text{Log} (z(z-1))$ are $z = 0$ and $z = 1$. A branch transition does not occur when a complete circuit is made about both branch points. It's only when the circuit encloses a single branch point that a branch transition occurs.