Homework 2 due Friday, October 11 at 5 PM.

Transcription

1 Complex Riemann Surfaces Monday, October 07, :01 PM Homework 2 due Friday, October 11 at 5 PM. How should one choose the specific branches to define the Riemann surface? It is a subjective choice, but there are some that are more convenient to work with than others. Two approaches: Geometrically choose where the branch cuts are desired and then find the corresponding values of the angles to define the cuts. Simply make simple choices for the angles for the cuts and then deduce where the cuts are geometrically. To work from the desired picture for the branch cut: The natural choices for choosing the values of the angles at which to cut is that should be some multiple of, bearing in mind that the above branch cut must rely on some sort of cancellation of branch cuts associated to each branch point. And then experiment with these choices: A natural first try is When two branch cuts overlap (as here on the ray lying along the positive real axis:, the discontinuities might cancel so that there is no real branch cut. Let's check: ComplexAnalysis Page 1

2 The double branch cut corresponds to, where the angles jump to when the branch cut is crossed. So the jump in the angle as this double branch cut is crossed is and therefore the argument of F(z) changes by, so its value doesn't actually change across the jump, i.e., F(z) so defined is continuous across the double branch cut so it isn't actually a true branch cut. So we can actually define a branch of the function as follows: Comment: This turns out to be the same branch that would be defined if we had chosen the branch cuts to be at Let's find another branch with the same branch cut. The typical strategy is simply to shift the definition of the angle domains by some multiple of (so that one is simply making a different choice of angle, but the branch cuts are identical. For example: ComplexAnalysis Page 2

3 Notice that with this definition, every value of z will be associated to some values for the first branch, and the values for the second branch. Therefore the argument of F 2 (z) is shifted by relative to F 1 (z), so. This is not a general relationship between branches for all multivalued functions, just functions like this. Since there are only two branches, and we know that we must change branches at a branch cut, we conclude that the two branches defined above are glued to each other along the branch cut on the real axis within the interval [-1,1] and when this branch cut is crossed, one switches between the two branches. This is topologically the same Riemann surface as for (think Riemann sphere). Example with compounded branch cuts: Look for branch points. Clearly the inner square root has singularities at which implies. The log function will only have singularities when it is evaluated at. What values of z correspond to potential singularities of log? ComplexAnalysis Page 3

4 So the only possible branch points are at -1,1,13/12 Are they actually branch points? Following a small loop around z=-1 (or z=+1) which change the argument of the quantity inside of the square root by, so the square root itself will have a different value and the rest of the function (adding 5 and taking the log) does not collapse the two values. (there is no identity Therefore are actual branch points. What about z=13/12? Take a small loop around z=13/12. ComplexAnalysis Page 4

5 So on this loop, Notice that, depending on the branch of the square root, this either looks like evaluating log along a small circle about the value 10 or the value 0. Log doesn't have any singularity at 10, so following it about a small circle in the vicinity of 10 does not create any discontinuity. But following log about a small circle encircling the origin does create a discontinuity because the circle is traversed exactly once and 0 is a branch point for log. What this means is that 13/12 is a branch point for f(z) only on the part of the Riemann surface corresponding to one of the branches of the square root. Therefore different parts of the Riemann surface will have branch points appearing in one of the two following ways: ComplexAnalysis Page 5

6 Let's just to find our way make some concrete choices for angles to define the branch cuts and then see what geometric structures they correspond to. Express variables in polar form with respect to their branch points. We will just try to define a branch by making arbitrary cutting choices, and see what it produces. Out of lack of creativity, let's just make the same choices as one makes for defining principal branches This somewhat implicitly defines a branch of f(z) as follows: ComplexAnalysis Page 6

7 This works fine as a branch. But what is the branch cut structure in the complex plane? Note that the cut of the angle is not yet very explicit in terms of z. Let's make it explicit. The only way to get a solution is for But our choice of branch forbids this; it is impossible for So this putative branch cut isn't actually there on this branch! ComplexAnalysis Page 7