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Physics 834: Problem Set #2
Here are some hints, suggestions, and comments on the assignment.
Remember to keep track of the amount of time you spend doing the
(entire) assignment and record this number on your problem solution.
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.
- Functions in the complex plane
- A general strategy for finding solutions to zm=a
(where m is an integer) is to represent a as
rei(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.
- 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 = eiz 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.
- Again, the trig identities from Arfken make this straightforward.
Remember that cosh2 - sinh2 = 1 and
cos2 + sin2 = 1.
- The function w = 1/sqrt(z).
- Work in polar representation of z.
You'll have sign differences for the real and imaginary parts of
w on the different branches.
- The branches correspond to how many times you need to increase
θ through 2π until w repeats. That is, with
rei(θ+2πn), how many n's do
you need to consider.
- 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.)
- Small amplitude waves in a plasma.
- 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?
- 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.
- Cauchy-Riemann relations.
- Follow the example from class (page 26).
- 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).
- 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.
- 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.)
- 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).
- 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!).
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