Tuesday (xhour):
 Study: Sections 2.1 and 2.2
 Do:
 In Section 2.1: 10, 12, 13.
 In Section 2.2: 5, 11de, 15, 22, 25bde. (The answer in the
book to 11d is incorrect. In 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 authors 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 would be acceptable
here.)
 Suggested:
 In Section 2.1: 1ace, 3d, 5, 7, 8, 9.
 In Section 2.2: 4, 6, 12.

Wednesday:
 Study: Read Sections 2.3 and 2.4. We will not finish
section 2.4 today so I moved some of the homework from section 2.4
to Friday.
 Do:
 In Section 2.3: 4a, 12, 16.
 In Section 2.4: 3, 5.
 Suggested:
 In Section 2.3: 1, 3, 8, 11egf, 13, 14.
 In Section 2.4: 1, 2, 4 (from lecture), 6.

Friday:
 Study: Read Section 2.5 and 3.1. We will only cover a
small part of section 3.1 in lecture, but you should skip the
parts we don not talk about.
 Do:
 In Section 2.4: 8, 12, 14.
 In Section 2.5: 5, 6, 8, 18, 20*, 21*. (See below for hints
on 20 and 21.)
 Compare the asserions in problems 20 and 21. Why is there no
contradiction here?
 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 will 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 $z\mapsto
\ln(z)+i\operatorname{Arg}(z)$ 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 $\operatorname{Arg}
(x+ i y)$ is harmonic by computing its partials using inverse trig
functions and taking care to note what quadrant you are 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 we will agree to make this problem
less messy by making the above assumptions.)
 Suggested: In Section 2.5: 1b, 2, 3cd.
