Physics 7701: Problem Set #1

Recent changes to this page:

• 21-Aug-2013 --- Original version.

1. Practice with δ_ij and ε_ijk
   1. The implicit assumption here is that the vector components all commute, so that A_iB_j = B_jA_i, but you should state this assumption explicitly.
   2. It is easiest to first manipulate the expressions with two cross products, because we can use the "very useful identity" from the δ-ε handout that eliminates two ε tensors in favor of Kronecker δ's. Be careful about moving the del's when doing this: in this case the implicit assumption is that both a and b depend on x (but that a and b commute). Note that when you simplify one of the terms, the other follows immediately by switching all a's and b's. Try combining the two terms while you still have it in indices, remembering the product rule for derivatives!
   3. Similar manipulations. Here the magnetic moment is constant, so what does that imply about derivatives of it?
2. 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 25 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 solutions when you solve for z, remembering that logarithms have an infinite number of branches.
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. Why 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.
   2. 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. (Bonus) As noted above, the trig identities from Arfken make this straightforward. Remember that cosh² - sinh² = 1 and cos² + sin² = 1.
7. (Bonus) 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).
8. (Bonus) 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.)