Physics 834: Problem Set #2

Recent changes to this page:

• 04-Oct-2011 --- expansion of comments based on questions received
• 01-Oct-2011 --- clarification of z².
• 30-Sep-2011 --- Added references to Mathematica notebooks.
• 29-Sep-2011 --- Original version.

1. Functions in the complex plane
   1. A general strategy for finding solutions to z^m=a (where m is an integer) is to represent a as re^i(θ+2πn). Then you can raise this to the 1/m power and keep enough n's until you get repeat answers for z. Check the example from class (page 24 in the notes) and Lea section 2.1.2. The Mathematica notebook Complex Roots can help to check your results.
   2. You will find the trig identities in problems 6.1.10 and 6.1.11 in Arfken helpful here and elsewhere in this problem set. You should prove those that you use (it is sufficient to assume that you can directly generalize well-known formulas such as for sin(A+B) to complex z). The Mathematica notebook Complex Trigonometry may be of use here. Use z = x + i y and identify real and imaginary parts of cos z, then equate to real and imaginary parts of 100 (why can you do this?). An alternative approach is to set u = e^iz and solve a quadratic equation. Be careful to get all solution when you solve for z, remembering that logarithms have an infinite number of branches.
   3. Again, the trig identities from Arfken make this straightforward. Remember that cosh² - sinh² = 1 and cos² + sin² = 1.
2. The function w = 1/sqrt(z).
   1. Work in polar representation of z. You'll have sign differences for the real and imaginary parts of w on the different branches.
   2. The branches correspond to how many times you need to increase θ through 2π until w repeats. That is, with re^i(θ+2πn), how many n's do you need to consider.
   3. For completeness, consider the image of the unit circle for each branch. (That is, first for θ from 0 to 2π, then from 2π to 4π, etc.)
3. Small amplitude waves in a plasma.
   1. When you assume that n, E, and v are proportional to exp(ikx-iωt), the constant in front will be complex in general. What can you cancel the exponential factor from the equations?
   2. Assume that the collision frequency is smaller than the plasma frequency. What is the signature of damping in the time dependence given by exp(-iωt)? Think about what happens in the other limit of large collision frequency, but you are not required to treat it.
4. Cauchy-Riemann relations.
   1. Follow the example from class (page 26).
   2. The first one here is (quite) tedious but straightforward given the trig identities from Arfken. (Note that z² means z times z and not the magnitude squared.) For the derivatives, you are supposed to check that du/dx + i dv/dx gives you the same answer as df/dz. The second one is reasonably straightforward. :) You can check your results with Mathematica. Try:
      ans1 = (x + I y)^2 Sin[x + I y] // TrigExpand
      u = Simplify[Re[ans1], Assumptions -> {Element[{x, y}, Reals]}]
      v = Simplify[Im[ans1], Assumptions -> {Element[{x, y}, Reals]}]
      D[u,x]
      D[v,y]
      D[u,x] - D[v,y]
      D[u, x] + I D[v, x] // Simplify
      Use the help to look us D, Re, etc. (it is often easiest to start with ?D).
   3. Just check the C-R relations. Writing w as a function of z requires that x and y only appear in the combination x+iy.
5. Taylor or Laurent series. The general strategy is to identify the non-analytic part at z0 and expand the rest in a Taylor series about z0, then combine. Here you expand the numerators. When expanding about z0, you may find it efficient to use w = z - z0, replacing z by w + z0, and then expand in w about 0. The radius of convergence will be determined by the nearest singularities outside the point specified. Is that point included in the region of convergence? For all of these, you can use the Mathematica Series command to check your answers. E.g., Series[Cos[z]/(z-1),{z,1,5}] will generate five terms of the expansion of the first problem (about z=1). (See the Mathematica notebook Complex Series for more examples.)
6. Section 2.6.3 of Lea has a nice summary of methods for finding residues. All you need is here (I used methods 1, 2, and 4 for the three problems here). You can generally check your result with Mathematica (see the notebook Finding Residues).
7. Basic applications of the residue theorem to calculate integrals. This is well documented in both Arfken (Chapter 7) and Lea (Chapter 2), and we'll do examples in class. For the first one, just apply the residue theorem --- there are no extra contours to consider. For the second one, be sure to comment why any extra parts of the integral (e.g., over a semicircle) should vanish (if you say they do!).