Physics 7701: Problem Set #2

Recent changes to this page:

• 01-Sep-2013 --- Added residue hint for 3(a).
• 29-Aug-2013 --- Added Mathematica hint for 2(a).
• 27-Aug-2013 --- Original version.

1. 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 two problems here and the other bonus problem). You can generally check your result with Mathematica (see the notebook Finding Residues).
2. Basic applications of the residue theorem to calculate integrals. This is well documented in Cahill (Chapter 5), 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!).
   You can use Mathematica to check the result for a particular contour if you parametrize the contour. For the circle in part a), the parametrization is in terms of theta:
   z = 2 Exp[I theta]
   and then the integral is found from
   Integrate[Cos[z]/z D[z,theta], {theta, 0, 2 Pi}]
   where D[z,theta] converts from dz to dtheta. Simplify the answer using FullSimplify.
3. Contour integrals
   1. Is it useful to extend the integral to minus infinity? Why can you do this? What kind of contour works for this one? (Does a semi-circle at large R vanish? If not, what are the alternatives?) To find the residue, note that the integrand satisfies the form of method 4. on page 28 of the Lecture 3 notes. This makes it quite easy!
   2. You could avoid the branch point here by changing variables to y = x^1/3, but what fun would that be? :) Instead, design your contour to avoid crossing a branch cut on the positive real axis and follow the steps as in the example done in class or similar ones in the texts. This problem is quite similar to Example 2.23 in Lea and you might use the solution there as a guide. Be careful to check for contributions from all the pieces (big circle, little circle, both sides of the branch cut) and remember that you'll get a different answer for the integration on top (θ = 0) and bottom (θ = 2π) of the cut. Also remember when calculating the residues that the branch cut was chosen so 0 < θ < 2π, so determine the angles of the poles consistent with this range. Mathematica gives the result for this integral without any tricks.
   3. We'd like to convert this to an integral around the entire unit circle (and not just half of it); how do we do that and justify it? This is then one of the standard integral types we considered and you get to practice finding poles and calculating residues. (You can use Mathematica to find the poles.) Mathematica gives the answer for the full integral directly.
4. Integral representation of step function
   1. Consider the two cases, t>0 and t<0, separately. In each case, choose a contour. Remember Jordan's lemma in deciding on what contour to pick. (Note that Jordan's lemma applies to closing the contour in the lower half-plane with a large semi-circle as well as the upper half-plane. Which half-plane you close in is determined by whether the exponential e^ikz has positive or negative k.) Where is the singularity (or singularities)? In Mathematica, you can get the t<0 case with:
      Integrate[Exp[I k t]/(k - I eps), {k, -Infinity, Infinity}, Assumptions -> {t < 0, eps > 0}]
      and similarly with t>0. You can take the limit of ε to zero in your head, or use the Mathematica Limit function (Limit[stuff,eps->0]).
5. Fresnel integrals
   What is an appropriate choice of f(z) that can get you both the sine and cosine integrals? An integral in the radial direction from the origin in the complex z plane can be parametrized as z = r e^iθ with a fixed θ and r going from 0 to the R. When considering the contribution from the curved part of the integration contour, what matters is the dependence on R. If there are parts that are pure phases (e^iα for some real α) --- do these affect whether the contribution blows up or dies off as R gets large? What about real exponentials? You can make a upper bound on the integral as it is done in the proof of Jordan's Lemma (Arfken 6th ed. pg. 467 or Lea pg. 140), leaving an integral you can do with the conclusion that the contribution vanishes as R goes to infinity.
6. Another residue problem. Same hints as for problem 1!
7. Proving an identity with residues
   This is the same type integral as 3(c), and the suggestions there apply here as well. Remember the binomial expansion (you can look it up in the index in Arfken), which tells you the coefficient of each term in (a + b)^m (with integer m). Can you integrate term by term? Which terms give non-zero contributions? (If you think about each term as being a one-term Laurent expansion, the answer is immediate; you can also use the residue formula for m-order poles.) For the double factorial expression, note equation (8.33c) in Arfken, which you can quote.
8. Atomic collision integral
   Look at Arfken Example 7.1.4 for a similar integral. You may find it useful to split up |p| > 1 into the two cases of positive and negative p (and with the |p| < 1 condition as well).