Physics 7701: Problem Set #3
Here are some hints, suggestions, and comments on the assignment.
Recent changes to this page:
 03Sep2013  Original version.
 08Sep2013  Added some hints for the bonus problem.
 Contour integrals
 As in the example in the class notes,
choose the height of the rectangular contour so that the top
edge (where z = x + i times height) is such that the integrand
is a multiple of the original integrand. Should the height
depend on a, b, both, or neither?
The conditions on a and b are important to ensure that the
right and left edges vanish as you take R to infinity;
make sure you see how that happens.
For the Mathematica evaluation, you want to tell it to assume
that b > Re(a) > 0. You can do this with the
Assumptions option to the Integrate command.
In this case, Assumptions > {b > Re[a] > 0} is what
you want. (After Assumptions is a minus sign and
a greaterthan sign, which together make an arrow.)
 If you use the definition of a principal value integral
we originally used in class (the average of the integrals with
k  iε and k + iε), then this
result follows quickly from your result from PS#2 and a very
similar result with the opposite sign for ε.
But you can also follow the texts or the lecture notes on
treating principal
value integrals.
To get Mathematica to do a principal value
integral, add the option PrincipalValue>True to
the Integrate function.
 Laguerre's equation
Follow the examples in Arfken or Lea or Cahill for the Frobenius
method (see the reading) or the lecture notes.
Regular at the origin means that it doesn't blow up at x=0.
We try a solution that is a Taylor expansion times x^{p}
with p to be determined (Arfken uses the variable k instead
of p). Substitute the series, doing the derivatives term
by term, and equate coefficients of the each term. The lowest power
of x determines p (you should find p=0 here),
and then you get relations between
coefficients of successively higher powers of x.
(Note that it is only the equation with the lowest power of x
that you solve for p. An equation for the next lowest power
may determine whether a_{1} is zero or a free parameter.)
You should
be able to identify the general term, but it is easiest to see if
you try the first few individually. To show that when α
is an integer you get a polynomial, you must show that the equations for the
coefficients of the series are zero after a finite number.
You should be able to check your general result against the
first few Laguerre polynomials, which you will find in quantum
mechanics books (or just Google it).
Note that the Frobenius method only gives you one p value;
we will discuss later how to find a second solution (which has
a logarithm and so is not regular at the origin).
 Bessel equation
Again, follow the examples in Arfken or Lea or Cahill for the Frobenius method
(see the reading) or the lecture 10 notes.
The same comments as for Laguerre's equation
apply here (in both cases you are expanding about 0).
A difference here is that you will find two possible values
of p, which you should consider separately.
You should find that a_{0} and a_{1}
are independent parameters (for one of the p values),
which means you obtain two solutions.
You should recognize
the series that you get in the two cases (once you pull out
an overall power of x). You can check it by noting
that the solutions are Bessel functions of order 1/2, which
you can find (for example) in Wikipedia under "Bessel function"
(search for spherical bessel functions).
 If you divide by x and then take the limit of x
to infinity, the equation with constant coefficients will be revealed.
Use e^{sx} as an ansatz and solve for s.
Then introduce v(x) as discussed in the lecture notes (or in the
texts) and solve the equation directly (you might try the
substitution w = dv/dx).
In the end you should find two independent solutions.
Mathematica can tell you the general solution as well as the solution
to the asymptotic equation (use DSolve as in the example notebooks
on the mathematica page). Note: it is sufficient to find the
solution for positive x.
 Here "simplify" means to eliminate the appearence of x^{2}
in the differential equation. After finding the equation for
v(x), it is Frobenius time again (actually you should find
ordinary power series for the two solutions).
 Langmuir waves

Landau says the integration contour should pass under the pole  this
is equivalent to pushing the pole upward by iepsilon.
Then use the magic identify in the integral.

You want to integrate by parts to move the derivative from f(v) to
(wkv)^{1}. Then you can Taylor expand the denominator
and integrate term by term.
Does the principle value matter?
 What would the sign of the imaginary part be in this case? Is it
physical?
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