Some Simple Examples of Riemann Surfaces
Let us start with some simple examples of the form where is a function whose branch points and branch cuts are easy to find and where it is easy to exclude them in graphics to get smooth surfaces. Because we are interested in the visualization of the Riemann surfaces we will from time to time explicitly exclude branch points, branch cuts, and poles, and start a distance away from such points with the polygons of the surface (, say for graphical purposes). This is also a good idea from the point of view that we are dealing mainly with logarithmic branch points (this means at the same time logarithmic singularities), and so the pictures have to be cut off vertically anyway.
Because later on we often refer to the location of the branch cuts of various functions, let us give a listing of the location of the branch cuts of the most basic transcendental functions. The next section deals with the inverse trigonometric functions and inverse hyperbolic functions in greater detail.
Table 1. Location of branch cuts in some transcendental functions.
The last two columns of Table 1 list the magnitude of the jump of the real and imaginary part across the branch cuts. The expressions marked as complicated for
The other sheets are shown below. As discussed previously, and from the Weierstrass methods dealt with in the next installment, we know that the other sheets arise from copies of the first sheet with multiples of added.
Sometimes one writes where Ln denotes the logarithm function only on the principal branch. By using this representation in a polar coordinate system, we can obtain the following picture representing the Riemann surface topologically as the function.
For representing Riemann surfaces in the following we mainly use the real or imaginary part. Of course, one could use other functions, such as the difference between the imaginary and the real part. (The difference between the real and imaginary part exhibits the logarithmic singularity at a bit better.)
Often it is hard to look at the inner parts of Riemann surfaces with more than two or three sheets. One workaround is to cut holes in the sheets to have a better look. We define a function
The next graphic shows the various types of polygons with holes on a sphere and illustrates the naming of the possible option settings for
To cut holes in the polygons of
Our next example is the function . Here is the principal sheet.
We see two branch points at ±1. Both are algebraic branch points of order 1. (This means they connect two sheets.) This can easily be seen by investigating the first terms of the Puiseux series expansion of around .
Using the periodicity of the function we can construct other sheets. The other sheets of follow from:
1. The fact that the solution of is given by . This is related to the logarithmic branch point of at . The series expansion of around shows this branch point clearly.
2. The fact that the
In the next installment we will generate pictures for all
The next picture shows the function with a square root branch point at and a logarithmic branch point at .
Here some sheets of the Riemann surface are displayed. We use the sum of the real and the imaginary part of the function. The imaginary axis is excluded to avoid jumps in the resulting picture.
The related function has a square root branch point at and a logarithmic branch point at .
In Part I  we discussed the function . A similar function in the context of this part is . The branch cuts arise from the log function, and are located at lines parallel to the real axis:.
Because the branch cuts connect branch points at , a picture of the Riemann surface just shows disconnected surfaces and does not look interesting. A more interesting function is because the inversion brings the essential singularity from infinity to the origin. By the inversion principle, the branch cuts occur along circles touching the real axis. Here is the principal sheet.
Without the branch cuts the function would be just the function . This fact offers a straightforward way to make a picture of some of the sheets of by just adding the term to which results in the other sheets of the log function.
The next function we will look at is . The principal sheet shows a branch cut along the negative real axis arising from the logarithm in the equivalent representation .
Rewriting in the form shows that the branch cut is caused by the log function. This allows us to construct some of the other sheets.
Now here is the principal sheet from the function .
Using the analytic continuation of arctan (derived from the periodicity of tan) we can construct some of the other sheets too.
We now deal with a slightly more complicated example: .
We see a couple of branch cuts parallel to the imaginary axis.
we get the following for the location of the branch points:
Be aware that
Now let us determine the location of the branch cuts. Taking into account that is purely imaginary for real it follows that the branch cuts of arctan(tan()/3) in Mathematica are the intervals .
By excluding the branch cuts from the x-y region covered in the picture above, we can make a more appropriate picture of the function arctan(tan()/3). Here is the definition for "one sheet" of the Riemann surface of this function. translates the surface vertically.
This shows the principal sheet.
Taking into account the periodicity of the tan function we can display some sheets of our Riemann surface.
Cutting the last picture along the x-z-plane shows the connections betweens the sheets better.
The next example we will investigate is .
The trivial branch point (arising from the inner log function) is the origin, and the nontrivial branch point is given as the solution of 0. This means at . The branch cut is a straight line from the origin to . Using this fact we can construct the following Riemann surface.
Nesting more logs results in more complicated pictures. The following picture shows a cross section of where we use five sheets for each of the log functions. The violet lines indicate the positions of the branch points at , , and .
We obtain similarly complicated pictures for functions like , where is implicitly defined by :
Now let us look at the function . Here the principal sheet is shown. We use
At we see logarithmic singularities. Expressing the function in logarithms shows this clearly.
Here some more sheets are shown.
The next function we will look at is . Here is the principal sheet with its branch cuts.
The interesting cross-like branch cuts arise from . They extend from the origin in direction , ,, and outwards.
For displaying the imaginary part, we only need to carry out the analytic continuation of the term. The does not give any new contribution.
Here is the absolute value. Now the term is also continued.
The function gives a very interesting-looking picture.
Looking at the picture one would guess that there are two branch points at . Looking at the rewritten form of
one sees that there is a branch point at arising from the arguments of the square root. The outer logarithm never assumes its branch point singularity at 0:
The outer logarithm also never assumes its branch point singularity at :
But anyway, the log function contributes an additional singular point at . Here the real and the imaginary parts of along the real axis are shown.
Interestingly at there is an immediate "jump" onto the branch cut of the log function without ever passing the corresponding branch point. The branch cut of log extends from to 0. The argument of log assumes the value at of the first sheet of and the value 0 at of the other sheet . So the branch cut runs from to and then back to .
Here are a few sheets of the Riemann surface of arccosh.
If instead the argument of arccosh is , we get a more complicated pattern of branch cuts.
Here are some sheets of the corresponding Riemann surface.
Now let us treat the function . The principal sheet shows branch cuts in the intervals .
Here some sheets of the Riemann surface are shown.
The "opposite" function is easier to describe--one just analytically continues the log part--but it looks more intricate pictorially because of the essential singularity at (arising from the essential singularity of at ).
The "bad" function has an accumulation point of singularities at , so the next picture becomes very weird around the origin.
We have finished showing Riemann surfaces of functions with simple straight line arrangements of branch cuts. Before treating the general case let us discuss the analytic continuation of the logarithm function from a Weierstrass point of view.
Converted by Mathematica May 9, 2000