14—Complex Variables 43514.7 Branch Points
Before looking at any more uses of the residue theorem, I have to return the the subject of branch points. They
are another type of singularity that an analytic function can have after poles and essential singularities.
√
z
provides the prototype.
Thedefinitionof the word function, as in section12.1, requires that it be single-valued. The function
√
z
stubbornly refuses to conform to this. You can get around this in several ways: First, ignore it. Second, change
the definition of “function” to allow it to be multiple-valued. Third, change the domain of the function.
You know I’m not going to ignore it. Changing the definition is not very fruitful. The third way was
pioneered by Riemann and is the right way to go.
The complex plane provides a geometric picture of complex numbers, but when you try to handle square
roots it becomes a hindrance. It isn’t adequate for the task. There are several ways to develop the proper
extension, and I’ll show a couple of them. The first is a sort of algebraic way, and the second is a geometric
interpretation of the first way. There are other, even more general methods, leading into the theory of Riemann
Surfaces and their topological structure, but I won’t go into those.
Pick a base point, sayz 0 = 1, from which to start. This will be a kind of fiduciary point near which Iknow
the values of the function. Every other point needs to be referred to this base point. If I state that the square
root ofz 0 is one, then I haven’t run into trouble yet. Take another pointz=reiθand try to figure out the square
root there. √
z=
√
reiθ=√
r eiθ/^2 or√
z=√
rei(θ+2π)=√
r eiθ/^2 eiπThe key question:How did I get fromz 0 toz?What was the path from the starting point toz?
0 1 2 −^3
z z
z zIn the picture,zappears to be at about 1. 5 e^0.^6 ior so.
On the path labelled 0, the angleθstarts at zero atz 0 and increases to 0.6 radians, so
√
r eiθ/^2 varies continuously
from 1 to about 1. 25 e^0.^3 i.
On path labeled 1, angleθagain starts at zero and increases to 0 .6 + 2π, so
√
r eiθ/^2 varies continuously from 1
to about 1. 25 e(π+0.3)i, which is minus the result along path #0.