Monday:
 Summary: Each class period will start with a short
summary of the lecture for that day. I will keep the meeting
open for the remainder of the class period to field questions.
Feel free to treat this time just like office hours. In
partiular, you should feel free to drop in and out of the
meetin if you desire.
 Homework solutions: Here
are selected solutions for the
first week's homework.
 Lecture: Watch both parts of Lecture 4. This material
is supplemented by sections 2.1 and 2.2 in the text.
 Do:
 Download the practice
document, answer the questions, scan your solutions to a PDF
document, and then upload your completed version via the canvas
page by 10:10 am on Wednesday (April 8th).
 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 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 would be acceptable
here.)
 Suggested:
 In Section 2.1: 1ace, 3d, 5, 7, 8, 9.
 In Section 2.2: 4, 6, 12.

Wednesday:
 Lecture: Watch both parts of Lecture 5. This material
is supplemented by sections 2.3 and 2.4 in the text.
 Do:
 In Section 2.3: 4a, 12, 16.
 In Section 2.4: 3, 5, 8, 12, 14.
 Suggested:
 In Section 2.3: 1, 3, 8, 11egf, 13, 14.
 In Section 2.4: 1, 2, 4 (from lecture), 6.

Friday:
 Lecture: Watch both parts of Lecture 6. This material
is supplemented by secions 2.5 and parts of section 3.1. We will
cover only a small part of section 3.1 in the lecture, so you
should have a closed look at the entire section on your own.
 Do:
 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'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 $z\mapsto
\ln(z)+i\operatorname{Arg}(z)$ 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'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.)
 Suggested: In Section 2.5: 1b, 2, 3cd, 10. Yikes!
Problem 10 is ridiculously complicated. Don't try it!
