**Instructor:****
****Jeffrey
Diller (click for contact info, list of my papers, etc.)**

**Official
Time and place:** Tuesdays and
Thursdays 12:45-2 PM in Hayes-Healy 117.

**Office
hours: **Starting Wed August 19 I’ll
hold regular office hours Wed from 4-6. My plan is to send you all an
invite to a recurring zoom mtg that starts 4 on Wed. If no one shows
by 4:30, I’ll shut it down, so please email if you plan to come
later than that, and I can restart the mtg if needs be.

**Textbook: ***Function
Theory of One Complex Variable (3*^{rd}*
edition) *by Robert E. Greene and
Steven G. Krantz. No single textbook can be expected to present all
points of view on this subject, so if you don't find Greene and
Krantz helpful, I'd suggest looking at these two reserve books, or
browsing the library stacks for math books whose numbers begin with
QA331. I’ll point you to three other sources in particular:

*Complex Analysis*by Lars Ahlfors. Ahlfors was one of the greatest complex analysts of the twentieth century, and his book is a classic, albeit maybe slightly dated by now.*Complex Analysis*by Stein and Shakarchi. This was written for an undergraduate course at Princeton, part of a four course sequence in analysis, but it’s really at the graduate level and quite good.Course notes by Terry Tao, who is arguably the world’s best living analyst. His account of things is almost always interesting and worth reading. Beware, however, that since the notes are in blog form with last-written entries appearing at the top, you have to scroll down to the bottom to get to the first “chapter.”

Turns out we’re not doing course reserves this semester, so I’m working on getting the libraries to get electronic access to the first two books.

**What is complex analysis? “**Calculus
meets complex numbers” might serve as a starting description of
complex analysis, but this doesn't do justice to the potency of the
combination. The notion of “imaginary” number has been around
since at least the Renaissance. But systematic attempts to take it
seriously and to integrate it into algebra, analysis, and geometry
only really got going in the nineteenth century with the work of
Cauchy, Riemann and others. Many facts (e.g. the prime number
theorem) that ostensibly belong to other areas of mathematics are
difficult, if not impossible, to state or prove without complex
analysis. And many physical theories (e.g. signal processing, quantum
mechanics) are most naturally expressed in terms of complex analysis.
In the first term of this two semester sequence, I hope to present a
large part of the ``classical (i.e. 19th century) theory'' of complex
analysis.

**What this course will cover: **Topics
for the first semester are fairly standard. I hope to cover chapters
1-7 of the textbook. A more precise list of topics, in roughly the
order we'll meet them, is as follows.

Geometry and arithmetic of complex numbers.

Definition and basic properties of complex analytic functions.

Contour integrals and Cauchy's Theorems.

Consequences and applications of Cauchy's Integral formula, including but not limited to

Liouville's theorem;

the maximum principle;

isolated singularities;

Calculus of residues;

The general form of Cauchy's theorems.

Conformal Mappings

Normal families and the Riemann mapping theorem.

(The Poincare metric)

(Schwarz-Christoffel transformations)

Harmonic Functions

(Subharmonic functions and the Dirichlet problem)

(Monodromy, Elliptic modular functions, and Picard's Theorems)

Parenthetic topics are things I'd like to cover if time permits. Time is, however, a rather unforgiving taskmaster.

**Homework: **Homework
problems will account for 50% of your grade in this course. I'll
assign new problems by noon every Friday and expect you to upload a
copy of your solutions by noon the following Friday. Note that we
might or might not grade all solutions. Regardless, I plan to at
least write up solutions to *all *the
problems and to make them available to you. I strongly encourage you
to collaborate with your fellow students when solving homework
problems, but you must write up solutions yourself. That is, you may
not copy from someone else’s solutions.

**Exams: **There
will be a midterm and final exam in this course. They'll be worth 20%
and 30% of your grade, respectively. Both will be in take home
format. The midterm will be given to you after class on Thursday Sept
17 to be completed in the next 48 hours. The final will be posted by
5 PM on 11/14/20. You will have until 5 PM 11/17 to complete it. You
are welcome to consult with me or the textbook or your class notes
for exams. Anything else (e.g. working with other students, looking
on the web or at other textbooks, etc) will be regarded as cheating.

**Necessary
Background: **Prior
exposure to complex analysis is helpful, but not necessary. Mostly
what I expect is familiarity with understanding and writing
mathematical proofs, particularly the epsilon/delta sort that arise
in undergraduate analysis (or advanced Calculus) courses. Familiarity
with topology of R^n (e.g. open, closed, compact, and connected sets
and the theorems concerned with them) is somewhere between helpful
and necessary.

*Any instance of cheating will be dealt
with according to Notre Dame’s
Academic Code of Honor.*

**Covid-19
stuff: **if
you’re like me, you’ve gotten way more email than a person can
absorb regarding steps we’re to take to mitigate the spread of
covid-19. If I see any problems in this direction concerning our
class, I’ll try to point them out. Please don’t hesitate to
return the favor if you see me doing something I shouldn’t or not
doing something I should or some such. Concerning specifics, let me
only note here that you’re supposed to pick a seat and stick with
it in the first week of class, reporting where you sat according
to these directions.