Physics 7701: Problem Set #2
Here are some hints, suggestions, and comments on the assignment.
Recent changes to this page:
 01Sep2013  Added residue hint for 3(a).
 29Aug2013  Added Mathematica hint for 2(a).
 27Aug2013  Original version.
 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).
 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.
 Contour integrals

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 semicircle
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!

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.
 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.
 Integral representation of step function
 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 halfplane with a large semicircle as well as the upper
halfplane. Which halfplane 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]).
 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.
 Another residue problem. Same hints as for problem 1!
 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 nonzero contributions? (If you think about each term as being
a oneterm Laurent expansion, the answer is immediate; you can
also use the residue formula for morder poles.)
For the double factorial expression, note equation (8.33c) in
Arfken, which you can quote.
 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).
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