WRITTEN HOMEWORK #8 SOLUTIONS
These problems are meant to give you practice computing zeros, poles, and residues
of a variety of functions. You should not turn them in, and the problems are entirely
optional, but it is worthwhile to get some practical cal
WRITTEN HOMEWORK #7 SOLUTIONS
(1) (a) Stein and Shakarchi, Exercise 12a, page 66. This problem asks you to
show that a harmonic function dened on the open unit disc is the real
part of a holomorphic function on that disc.
(b) Let u(x, y ) = log(x2 + y 2 )
WRITTEN HOMEWORK #6 SOLUTIONS
(1) Suppose f is holomorphic on an open set containing a rectangle R and its
interior, except possibly at a single point a in the interior of R. Suppose we
also know that f is bounded near a. Show that
f (z ) dz = 0.
R
This i
WRITTEN HOMEWORK #5 SOLUTIONS
(1) Let f (z ) be holomorphic on an open set and let z : [a, b] C be a C 1
function whose image is in . Show that f z : [a, b] C is dierentiable,
and that
d
(f z )(t) = f (z (t)z (t),
dt
for all a < t < b.
Solution. Let f (x
WRITTEN HOMEWORK #4 SOLUTIONS
(1) Let z0 be any nonzero complex number. Find the power series expansion for
1/z centered at z0 , and show that its radius of convergence is |z0 |. (Notice
that this is the largest the radius of convergence can possibly be,
WRITTEN HOMEWORK #3 SOLUTIONS
(This version of the assignment will have wording dierent than older versions
because I accidentally overwrote the original version. In the future I might change
this back to the original version.)
(1) Let k be a xed positive
WRITTEN HOMEWORK #2 SOLUTIONS
(1) Rigorously prove that an open disc in C is an open set. That is, if is an
open disc, and z is an arbitrary point of , show that there exists some
r > 0 such that Dr (z ) .
Solution. Let z = DR (z0 ) be any point in the op
WRITTEN HOMEWORK #1 SOLUTIONS
(1) For each of the following equations, give a geometric description of the set of
complex numbers (ie, describe how this set looks in the complex plane) which
solve that equation. The numbers z1 , z2 , . . . refer to arbitr
WRITTEN HOMEWORK #8 SOLUTIONS
(1) Stein and Shakarchi, Problem #15c, Chapter 3, page 106.
Solution. Let f (z ) = (z w1 )(z w2 ) . . . (z wn ). Since f (z ) is a polynomial,
it is an entire function. Furthermore, notice that |f (z )| is the product of the
WRITTEN HOMEWORK #9 SOLUTIONS
(1) Compute the following integrals.
1
(a)
dz , where C is the circle |z | = 4.
2
C z (z 2)
1
(b)
dz , where C is the circle |z | = 1.
2
C z (z 2)
ez
dz , where C is the square with vertices 2 2i.
(c)
2
C z +1
Solution.
(a) T