Note the particular way we will analyze Riemann surfaces is as described in Ablowitz and Fokas Sec. 2.3 (online)

Homework 2 due Friday, October 11

From last time, recall that a coherent strategy for working with multivalued functions is to introduce a Riemann surface as a domain for the multivalued function. A Riemann surface can be usefully visualized as a surface that projects in a many-to-one way onto the (extended) complex plane. Closed loops in the extended complex plane that enclose branch points might not correspond to closed loops on the Riemann surface, which resolves the problem of how the value of the function can change from the beginning to the end of a closed loop returning to its initial point. Non-closed loops on the Riemann surface can project onto closed loops of the (extended) complex plane. To work analytically with a Riemann surface, one typically defines branches which are subdomains of the Riemann surface which are in a 1-1 correspondence with the (extended) complex plane (except perhaps for some branch cuts).

For the case of logarithm (which is somewhat generic), we can think of the Riemann surface as parameterized in a 1-1 way by polar coordinates:

This is a many-to-one parameterization of the complex plane because the points correspond (project) to the same point on the complex plane.

A branch would correspond to specifying a subdomain such as:

Which involves a branch cut along the ray:

The branch cuts are simply artificial discontinuities introduced by the choice of which subdomain of the Riemann surface you want to put in 1-1 correspondence with the (extended) complex plane.

Let's now consider other multi-valued functions: For some complex constant We define this function to be:

Riemann SurfacesThursday, October 03, 20132:01 PM

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For the case when is a real integer, this turns out to be a single-valued function (so the Riemann surface is just the (extended) complex plane) because the multivaluedness of log gets projected out by the subsequent exponentiation:

•

(corresponds with real-valued definition on the positive real axis). This generally speaking is a multivalued function in the same way that log is, with the following variations.

Real rational values •

This is a multivalued function, but with only q points on the Riemann surface corresponding to a given point on the (extended) complex plane, so the Riemann surface should be thought of having q sheets.

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One can define branches just as for logarithm by introducing a branch cut. Again the branch points are at 0 and (only place the function is singular, and one can check that going around small closed loops does not return one to the same function value.) And so one can define branch cuts like:

Choose a branch by specifying a range of parameters for the argument about the branch point that has a 1-1 relationship with the complex plane:

And then defining the branch of the multivalued function as:

Besides defining branches, it is also important to describe how the branches are related to each other. In this example, we could take the following specific branches:

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The branches are connected to each other on the Riemann surface in that if one crosses a branch cut corresponding to branch j in a counterclockwise direction relative to the origin, then one moves to branch j+1 (mod q). Likewise if one crosses the branch cut in a clockwise direction (relative to the origin)then one moves down to branch j-1 (mod q). The branches for logarithm are similarly connected, except one drops the mod q. One can see from this construction that the fundamental group corresponding to the Riemann surface for zp/q is

Intermediate example:

We might expect multivaluedness arising from the square root.

z2-1 is analytic everywhere•z1/2 is analytic away from branch points at •

First let's identify the branch points. Think of f(z) as a composition of more basic functions:

Only possible singularities (therefore branch points) is where w(z) fails to be analytic (nowhere) or maps to a singularity of square root:

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These are the only possible branch points of f(z) ; everywhere else analytic.

But are they actually branch points?

Let's see by drawing small circles around each candidate and seeing how f(z) behaves as one traverses these circles in the complex plane.

Note we are measuring as an argument with respect to 1 ,rather than the origin in this parameterization.

arg (z-1) changes by •arg (z+1) does not change.•arg w(z) = arg (z2-1) = arg (z+1) + arg (z-1) changes by 2 •The subsequent square root operation will therefore have its argument

change by

and therefore not return to the same value at the end

of the loop as at the beginning.

•

As we traverse this circle, we note that :

This shows that z=1 is a branch point, and a similar argument show that z=-1 is also a branch point.

Now let's consider a "small loop" around can be described by the parameterization:

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(alternatively, map

and then draw a small loop

around z'=0)

Let's look at w(z):

Therefore after passing through the square root, the argument of f(z)

would change by

which does give the same value as at the

beginning of the loop. So is not a branch point.

Therefore we have shown that the branch points for f(z) are precisely -1,1.

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Therefore we have shown that the branch points for f(z) are precisely -1,1.

And we know that the square root gives two possible values, so we'll have a Riemann surface with 2 sheets, fastened at branch points -1,1. To describe the Riemann surface in analytical terms, we'll want to introduce suitable branch cuts to define branches.

Let's take the branch cut on the left for simplicity. We define branches as follows:

Introduce polar coordinates with respect to each branch point:

We note that are not independent variables, but all that's important is that they are computable as functions of z, and the above polar coordinate description makes it (implicitly) clear how to do this.

Then one defines a branch corresponding to the branch cut by using it to restrict the range of angles (w.r.t. the branch points) to be uniquely defined for a given z.

Where:

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The branch cut we've drawn corresponds to:

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