Math 303 - Complex Variables
Extra Credit Homework due March 2
Question 1. Consider the following set in the complex plane:
Q+iQ={x+iy |x, y ∈Q}.
Thus, Q+iQis the set of all complex numbers where both the real part and imaginary part are rational
numbers.
(a) Is the set Q+iQopen, connected, bounded, a domain?
(b) What is the boundary of Q+iQ?
(c) Is Q+iQclosed?
Question 2. This question is in our textbook (Question 3.4.6). You may wish to consult the figure on
pg. 131 in computing your answer. This question asks you to find a general formula for the boundary
value problem for the annulus of inner radius r1and outer radius r2about the origin. The boundary
conditions are that your harmonic function φtakes on the value φ=Bon the inner circle and the
value φ=Aon the outer circle (see figure). We will solve this by using the general formula
φ(z) = CLog|z|+D
(a) In terms of the parameters r1, r2, A, and B, find the general solution φ(z) that solves this boundary
value problem.
(b) If we wanted to solve the boundary value problem for the disk or radius r2(and thus there is no
inner circle), we would simply take your answer from (a) and let r1→0 (since this would shrink
the inside hole to nothing). Do this. What is the function that is harmonic on the disk or radius
r2and takes on the value φ=Aon the boundary circle of radius r2?
(c) The textbook question asks “Now don’t you feel foolish?” Given the answer your obtained in (b),
explain this (admittedly pretentious) comment.
1
