Assignment 3 - Complex Analysis
MATH 440/508 – M.P. Lamoureux
Due Friday, Nov 6 at lecture
Note: Your work will be graded both on correctness of the mathematics, and
the clarity of presentation. Prove your statements, or justify your calculations, as appropriate.
These questions our from our textbook by Stein and Shakarchi. The text
includes some hints that might be helpful. My notes below indicate why I
think these particular problems are interesting.
1. Page 106, # 16. Continuity of zeros.
We know we can use Roche’s theorem to show that perturbing an analytic
function by a small amount does not change the number of zeros in a region.
This exercise asks you to prove that the location of the zero changes continuously with the perturbation. Thus, if you have a numerical algorithm to find
zeros of a function, you can have confidence that small errors in the input
function will only result in small errors in finding the zero of that function.
2. Page 106, # 19. Max modulus principle for harmonic functions.
You may have noticed from the Poisson kernel formula for a harmonic
function (last assignment) that in the special case at the centre of a circle of
radius ρ, we have
1 Z 2π
u(z0 + ρeit ) dt.
u(z0 ) =
2π 0
That is, the value of the harmonic function at the centre of a circle is equal
to the average of its values around the circle. So, you might expect that u
can’t get much bigger in the middle than on the sides. This problem asks
you to prove the max principle on more general regions.
3. Page 109, # 3. Laurent series expansions.
1
You may have noticed that a function f (z) = e1/z has an essential singularity at z = 0, so there is no way write it as f (z) = z −n g(z), where g
is analytic at zero. So we don’t immediately know how to write its series
expansion, from a Taylor series expansion of g. On the other hand, we know
how to expand ez , so it makes sense to write
e1/z = 1 +
1
1
1
1
+
+
+
+ · · · for z 6= 0.
2
3
z 2!z
3!z
4!z 4
Laurent series gives you a general way of computing such infinite series, that
have both positive and negative terms for z.
Note: We see that e1/z will have a residue of 1 at z = 0. However, some
of our theorems (e.g. principle of the argument) only work for meromorphic
functions, so you might like to consider what theorems fails for such a function
like e1/z .
M h(x)
dx.
4. Page 110, # 5. The function g(z) = −M
x−z
Such functions come up in inverse problems in electrical engineering, for
instance. The function h(x) along the line [−M, M ] might represent electric
charge along a strip of an insulator or semiconductor, while g(z) is the electric
field you measure nearby. If you measure g(z) at several points, can you
deduce what h(x) is?
R
5. Page 132, # 2.
This is actually a practical way to solve an inhomogeneous linear differential equation. The formula
u(t) =
Z
L
e2πizt ˆ
f (z) dz
P (z)
can be computed numerically, and gives a nice approximation to the solution
to the DE.
5. Page 133, # 2.
This “three line lemma” is related to KMS states in quantum statistical
mechanics (Kubo-Martin-Schwinger states).
2
Or, more concretely, recall we saw functions like f (z) = e−iz which are
bounded along the positive real axis (x > 0) and the positive imaginary
axis (y > 0) but they are not bounded in the upper quadrant. In fact, it is
unbounded along the third line z = (1 + i)t, t > 0. The three line lemma
says, if the lines are in a strip, this unboundedness can’t happen.
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