Math 220A - Complex Analysis
Welcome to Math 220a!
This course provides an introduction to complex analysis. We
will cover holomorphic and meromorphic functions, Taylor and Laurent
theorem and its applications, calculus of residues, the argument
principle, harmonic functions, sequences of
holomorphic functions, infinite products, Weierstrass factorization
theorem, Mittag-Leffler theorem.
Math 220 is a three quarter sequence. Adrian Ioana will teach Math 220b
and Math 220c in Winter and Spring.
Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu,
Lectures: WF, 11:00-12:20, AP&M 5-402.
- Wednesday 1-2 in AP&M 6-101.
Textbook: Complex Analysis by Lars Ahlfors.
Course Assistant: Pun Wai Tong, p1tong at ucsd dot edu, APM 6-331.
Office hour: Tuesday 11-1.
lecture or by appointment. Also, feel free to drop in if you see me in my
- The minimal prerequesites are Math 140
A-B. However, this is a
graduate level course, so at times, we may use notions from related
including topology and real analysis. I am happy to discuss prerequisites
on an individual basis.
If you are unsure, please don't hesitate to contact me.
- There will be a Midterm on November 4, in class,
as well as a Final Exam on December 8, 11:30-2:30. The final grade will be
based the following scheme:
20% homeworks and regular attendance of
lectures, 30% Midterm, 50 % Final Exam.
The problem sets are mandatory and are a very important part of the
course. The problem sets are due at 4pm in the TA's mailbox.
- First class: Friday, September 25.
- Midterm Exam: Wednesday, November 4.
- Veterans Day: Wednesday, November 11.
- Thanksgiving break: November 26-27.
- Last class: December 4.
- Final Exam: December 8, 11:30-2:30.
There will be no lectures on Oct 28 and Oct 30. Make-up
lectures will be scheduled on Monday, October 12 and Monday, October 19,
at the usual time.
- There will be no lecture on Wednesday, Nov 25, the day before
Thanksgiving. The make-up lecture is scheduled for Monday, Nov 23.
- Lecture 1: Overview and motivation. Complex differentiable
functions. Comparison between complex and real
differentiablity. Cauchy-Riemann equations. Harmonic functions.
- Lecture 2: Complex differentiability and conformal maps.
Power series and radius of convergence. Power series are complex
differentiable. Complex exponential, sine and cosine.
- Lecture 3: Branches of the logarithm. Principal branch. nth
Integration over continuously differentiable paths.
Basic properties and
examples. Some results about existence of primitives.
- Lecture 4: Goursat's lemma. Existence of primitives of complex
differentiable functions defined over discs. Primitives along continuous
paths and integration over
- Lecture 5: Winding
number. Homotopies. Invariance of the integral of holomorphic functions
over homotopic paths. Existence of primitives and branches of the
logarithm in simply connected sets.
- Lecture 6: Primitives along homotopies. Integrals
functions depend on the homotopy class of
paths/loops. Enhancements. Cauchy's
- Lecture 7: Differentiable functions and Taylor series.
Cauchy's integral formula for
estimates. Liouville's theorem. Fundamental theorem of algebra. Zeroes of
- Lecture 8: Inverse function theorem. Open mapping theorem.
Maximum modulus principle. Laurent series. Singularities and their type.
Removable singularity theorem.
- Lecture 9: Poles. Meromorphic
functions. Essential singularities. Casorati-Weierstrass theorem. Residues
and methods to compute
them. Statement of the residue
theorem and connections with Cauchy's formula.
- Lecture 10: The residue theorem. Proof
for Jordan curves. Cauchy's theorem
for chains which are homologous to zero via Liouville's theorem. The
proof of the residue theorem for arbitrary curves.
- Lecture 11: Residues at infinity and sum of residues over the
Riemann sphere. Applications to real analysis: trigonometric
integrals, rational functions, examples from Fourier analysis.
- Lecture 12: More examples of integrals from Fourier analysis.
Logarithmic integrals. Mellin transform.
- Lecture 13: The argument principle. Applications to elliptic
functions: matching the number of zeroes and poles, and the sum of zeroes
and poles. Rouche's theorem and applications.
- Lecture 14: Sequences and series of holomorphic functions.
Local uniform convergence. Weierstrass
convergence theorem. Hurwitz's theorem. The Weierstrass problem and motivation
for infinite products.
- Lecture 15: Convergence and absolute convergence of infinite
products. Infinite products of holomorphic functions. Examples.
Logarithmic derivative of infinite products.
Product expansion for the sine function.
- Lecture 16: Weierstrass elementary factors. Weierstrass
factorization of entire functions. Genus. Meromorphic functions as
quotients. Reinterpretation in the language of divisors.
- Lecture 17: Examples of Weierstrass factorization. The product
expansion of the gamma function and various identities. Euler's constant. Weierstrass problem for arbitrary
- Lecture 18: Mittag-Leffler theorem.
Meromorphic functions with prescribed poles and principal parts.
Examples: Weierstrass elliptic functions. Construction of entire
functions with prescribed values.
Homework 1 due
Wednesday, October 7 - PDF
Homework 2 due Wednesday, October 14 - PDF
Homework 3 due Wednesday, October 21 - PDF
Practice problems for the midterm - PDF
Midterm - PDF
Homework 4 due Friday, November 13 - PDF
Homework 5 due Monday, November 23 - PDF
Homework 6 due Wednesday, December 2 - PDF
Practice problems for the final - PDF.
Final - PDF