From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: basic question
Date: 11 May 1999 20:26:17 GMT
Newsgroups: sci.math
Keywords: What are Riemann surfaces?

In article <7gu86k$n3a$1@nnrp1.deja.com>,  <dsj@engrs.unl.edu> wrote:

>I am very new to complex analysis. I was wondering if any one could explain
>clearly the concept of branch cuts and Reimann Surfaces

What does the graph of  y=sqrt(x^4-x^2)  look like? Your calculator will
show you two disjoint curves, but in some sense it is wrong. In order
to graph the function, it must select one of the two possible square roots
for each  x. It has followed the convention of selecting the positive root
in each case. Yet it is difficult to extend this convention to cover other
cases. You see, you have graphed a portion of the curve described implicitly
by  y^2=x^4-x^2. What if I gave a more subtle curve like 
y^2+y/10 = x^4-x^2 ?  y^3+7y=x^4-x*y ? ... You soon realize that for each
x you are to pick one of several roots  y, and there isn't really a
convention to guide you in your selection.

Now consider doing all this over the complex numbers. Choose one solution  y0
of the equation for one value  x0  of x. I'll take  x0=2, say, and then pick
y0=+sqrt(12) (although y0=-sqrt(12) is OK too). Clearly, a reasonable graph of
the rest of the function would have  y  vary continuously with  x  away from
this one point. The process of "analytic continuation" guarantees that
you will be able to do just this near most choices of (x0,y0) -- remember,
the complex numbers are algebraically closed, so for _any_ x there will be
the right number of y's to go with it (counted according to multiplicity,
assuming a monic polynomial in  y). That is, now there is no "gap" between
"two parts" of the curve  y=sqrt(x^4-x^2); you go on your merry way, say,
tracing out a semicircle from x=2 through x=2i and over to x=-2, at each
point adjusting  y  a bit to get a solution to  y^2=x^4-x^2.

The surprise here is that the by the time you get over to x=-2, the "right"
value of  y  is not the positive square root but the negative one, that
is, it's actually "more right" to write  y = x sqrt(x^2-1).

Now, if you try to keep this up, following this big circle, you come
back to  x=2  with the positive square root again, so that you've linked
together the two branches in a consistent way.

On the other hand, if you try to compute a value of  y  for  x=1/2, say,
(this  y  will be purely imaginary of course), then you get one of the
two square roots if you follow a small counterclockwise circle from x=2.
But if you follow a small _clockwise_ circle from the same starting point,
you are led to the _other_ square root. Equivalently, if you follow a
small circle around from x=2 to x=1/2 and back to x=2, you come back
with the "wrong" value of  y.

The difference between this set of computations and the earlier one is that
the square root function behaves badly near zero, and so sqrt(x^4-x^2)
is rather nastily behaved around both x=1 and x=-1. This nastiness shows up
as soon as you try to wrap all the way around either of these points.
As it turns out, though, the two nasty bits "cancel out", so that any circle
which wraps around both x=1 AND x=-1 will not exhibit this bad behaviour.
You can use this analytic continuation to define  sqrt(x^4-x^2)  at any
point, in a consistent way, if you agree never to move along a path which 
encircles one but not both of these two points.

Here's a way to do this: pick a path in the complex plane joining  x=-1
and x=+1. View it as a barrier, that is, use analytic continuation to 
extend your domain of definition along any path from x=2 to any other point,
but don't let your path cross the barrier. If you agree to this procedure,
you'll get a consistent definition of  sqrt(x^4-x^2)  everywhere except
right on the barrier.

If you could "graph" this function in  C x C, you'd get something which is
a (real) 2-dimensional surface, with one point attached to every point
of the complex plane except for the points in your barrier. Viewed from
above, then, this graph is just (a wavy form of) the slit complex plane itself.



Of course, you can begin the whole process with the OTHER solution to
y^2=x^4-x^2 at x0=2. As you might imagine, the result of analytic continuation
will be that you'll have a different definition of  sqrt(x^4-x^2) across
most of the complex plane; the values now will be exactly the negatives
of what they were before. The graph of this function would then be a 
second wavy copy of the slit complex plane.

Now, you may recall that those opposite values almost crept in to the
definition of  sqrt(x^4-x^2)  the first time: I got the two different
square roots at  x=1/2, say, by following different paths from x=2.
More precisely, for each value of  x  on the barrier you drew, there are two
possible values of  y: one shows up when you think of  x  as being one
one side of the barrier in the first construction, or when you think of
x  as being on the other side of the barrier in the second construction;
of course the other value of  y  showed up on the "other" side in each case.


So here's what we decide to do: we make the "whole" graph of the function
by combining these two. That is, we look at ALL the points in  C x C  where
y^2 = x^4 - x^2. There are two such points for each  x, one in each of the
two graphs we've made. But the set of all these points fits together smoothly
to make a surface in C x C. As you might have guessed, the whole business
with the slits is just a fake, a convenient way of splitting the surface into
two parts we can see; but you can travel readily from one part to the next
in  CxC. Indeed, the analytic continuation game has already shown us what
to do: as we get near one side of the barrier in the first graph, we have
to jump to the other side of the corresponding point in the barrier in
the second graph.

So here's how you view this surface in  C x C: take two copies of the complex
plane, slit from x=-1 to x=+1 in each ,then glue opposite sides of the two
slits together.

Having trouble doing that? Be flexible! Don't just leave the top copy of  C
sitting there while you try to glue this. Rather, flip this copy of  C  over,
180 degrees, around the real axis. Now the sides of the slit in each copy
of  C  which must be glued together are right above each other! Stretch, 
glue, bend, twist, and soon you have a shape like a spool: each copy of  C
is one of the two parallel plates, the slit has been opened up and turned
into the central cylinder of the spool.

This final surface is almost the Riemann surface associated with the
equation  y^2=x^4-x^2. There is the little matter of points at infinity,
as well as natural questions you should be asking about smoothness and so on,
but the long and short of it is that this procedure will leave you with a
compact, oriented surface without boundary.


While I've got all this on the table, let me point out that some equations
are just naturally different from others. If you try this process with
y^2=x^2-1, you put a slit from x=-1 to x=+1 and do all the gluing only to
find at the end that you have a sphere. If you try  instead with
y^2=x^3-x, you find you must put barriers to stop curves from wrapping
around any single one of {x=0, x=1, x=-1}. The way to do this is to use
_two_ slits, say joining  x=-1 to x=0 and then one from x=0 to x=infinity.
This time all the contortions will leave you with a torus, not a sphere.
Likewise equations  y^2=x*(x-1)*(x-2)*...*(x-2n)  will produce surfaces with
n  handles. We say the surface has   genus  n,  and then likewise say the
equation (or the "complex curve") has  genus  n.  This genus  is an important
invariant of the Riemann surface; it says a lot about the behaviour of the
functions defined on the surface, for example.

I'll close with the observation that an integral of e.g. the form
integral( sqrt(x^3-x) dx )  can be thought of as the integral of  y dx
on the corresponding Riemann surface. Here  y  is just the second coordinate
of these points in  C x C, but it thus becomes a differentiable function
defined on the surface. Well, as I said, the genus of this surface is  1, 
and genus determines some basic things about what kinds of functions "live"
on the surface. So the integral in question is the integral of a function
on this surface of genus  1; it turns out, then, that its antiderivative
must also be, um, a genus-1-kind-of-function. Well, there's nothing all
_that_ exotic about these functions, except that they're not the same
as the genus-0-kinds-of-function, which includes all the trig and exponential
functions we call "elementary". It is in this way that we can decide that
"you can't do integral( sqrt(x^3-x) dx ) ", that is, that this function
has no elementary antiderivative.

dave

Riemann surfaces are part of complex analysis,
	index/30-XX.html
Symbolic integration is carried out more formally as a part of Field Theory,
	index/12-XX.html