**Instructor:****
****Jeffrey
Diller (click for contact info, general policies, etc.)**

**Official
Time and place:** MWF 9:35-10:25 in Debartolo 119.

**Textbook: ***Function
Theory of One Complex Variable (3*^{rd}*
edition) *by Robert E. Greene and
Steven G. Krantz. I have also placed some books on reserve in the
math library on the second floor of CCMB:

*Complex Analysis*by Lars Ahlfors.*Functions of One Complex Variable*by John Conway.*Real and Complex Analysis*by Walter Rudin.*An introduction to complex function theory*by Bruce Palka.*Complex Analysis: the Geometric Viewpoint*by Steven G. Krantz.*Theory of Complex Functions*by Reinhold Remmert.

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 some of the reserve books, or browsing the math library stacks for books whose numbers begin with QA331. Ahlfors was one of the greatest complex analysts of the twentieth century, and his book is a favorite among mathematicians. Conway's book is similar to Ahlfors but heavier on details, a feature many beginning students appreciate. Rudin's book integrates real and complex analysis into a single presentation. Palka's book is the most detailed of all the reserve books. It's also a good source for further problems to practice on. The reserve book by Krantz isn't really a text book at all, but an inspiring and award winning monograph on the use of geometric ideas in complex analysis. I'd suggest having a look at it once we're reach, say, the Riemann mapping theorem. Finally, the book by Remmert gives a history of the material we'll be covering in class.

**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 numbers'', shunned for its
apparent absurdity and invoked for its usefulness, has been around
since at least the Renaissance. But systematic attempts to take the
notion 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 and pick up your solutions to old ones almost
every Friday in class. Note that I don't intend to grade every last
problem---I prefer grading a couple of problems well to grading a
bunch of them haphazardly. At any rate, I plan to write up solutions
to *all *the
problems and to make them available to you.

**Exams: **There
will be a midterm and final exam in this course. They'll be worth 20%
and 30% of your grade, respectively. The midterm will take place ???
Wednesday, October 14 from 5-7 PM in ???, and the final on Tuesday,
December 15 from 8-10AM in ???.