Monday:
 Study: Read sections 2.1 and 2.1
 Do:
 To save me some typing, I am no longer
breaking out the "suggested" problems. Instead, interesting problems
that you should look at but not turn in will be listed in
parentheses.
 Section 2.1: (1ace, 3d, 5, 6ab, 7, 8, 9), 10, 12, 13.
 Section 2.2: (2,4), 5, (6), 11de, (12), 15, (18), 22, 25bde.
 Comment on Problem 15: We know from lecture that a complex
valued function is continuous if and only if its real and
imaginary parts are. Hence it is "legal" to use that in
homework. The author had in mind you proving one direction of
that in this problem. So you can either cite that result, or
try to prove it from the definitions. Either way is
acceptable here.
 Comment on section 2.2, \#11d. The answer in the back of
the book is incorrect.

Wednesday:
 Study: Read sections 2.3 and 2.4
 Do:
 Section 2.3: (1), (3), 4a, (8, 11efg), 12, (13, 14), 16.
 Section 2.4: (1, 2), 3, (4 mentioned in lecture), 5, (6), 8,
12, 14.

Friday:
 Study: Read sections 2.5 and 3.1. Section 3.1 is faily
long and we'll only briefly discuss it in lecture. So, you'll be
on your own there and should read that section carefully.
 Do:
 Section 2.5: (1b, 2, 3cd), 5, 6, 8, (10,) 18, 20* and 21*.
 Compare 20 and 21! Why is there no contradiction there?
 Problems 20 and 21 are a bit harder than usual. I've included
some hints below. But while I wouldn't call then "extra credit",
don't waste too much time on them if you're stuck.
 I didn't understand the author's hint for problem 20. Instead,
I used the Fundamental Theorem of Calculus. We want to show a function $u$ harmonic in $D=\{z\in
\mathbf{C}:zz_0< d\,\}$ has a harmonic conjugate in $D$. Then
let $z_0=x_0+iy_0$. Now if $a+ib\in D$, then the line seqments from
$a+ib$ to $a+iy_0$ and from $x_0+iy_0$ to $a+iy_0$ are also in $D$.
Define $$v(a,b)=\int_{y_0}^b
u_x(a,t)\,dt +\phi(a),$$
where $\phi$ is a function to be defined by you later. You may assume that we
know from our calculus courses that this defines a continuous
function $v$ with continuous second partial derivatives. Note
that the second term in the displayed equation above depends only
on $a$ and not on $b$. You
may also assume that $$\frac{\partial}{\partial x}\int_{y_0}^b
u_x(a,t)\,dt =\int_{y_0}^b u_{xx}(a,t)\,dt.$$
(This is called "differentiating under the intergral sign", and
we'll also assume this from calculus.)
 For 21, the idea is that any two harmonic conjugates in a
domain must differ by a real constant. You may assume without
proof that $z\mapsto \ln(z)$ is harmonic on
$\mathbf{C}\setminus\{0\}$ and that $z\mapsto
\ln(z)+\operatorname{Arg}(z)$ is analytic on the complement $D^*$
of the nonpositive real axis. (If you wish, you can check that
$\ln(\sqrt{x^2+y^2})$ is harmonic on $\mathbf{C}\setminus\{0\}$,
and you can show $Arg (x+ i y)$ is harmonic by computing its
partials using inverse trig functions and taking care to note what
quadrant you're in  but we will find a better way later. Then
the analticity of $\ln(z)+\operatorname{Arg}(z)$ follows from one
of our CauchyRiemann theorems. But let's make this problem less
messy by making the above assumptions.)
