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. Jim Agler and Kiran Kedlaya will
teach Math 220b
and Math 220c in Winter and Spring.
Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu,
Lectures: WF, 12:30-1:50, AP&M B-412.
- Wednesday 2-4 in AP&M 6-101.
Textbook: Complex Analysis by Lars Ahlfors.
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 2, 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 in class.
- First class: Friday, September 23.
- Midterm Exam: Wednesday, November 2.
- Veterans Day: Friday, November 11.
- Thanksgiving break: November 24-25.
- Last class: Friday, December 2.
- Final Exam: Thursday, December 8, 11:30-2:30.
- Office Hour for Week 2 has been moved to Tuesday, October 4, 2-4PM.
There will be no lecture on November 4. Make-up
lecture will be scheduled on Monday, October 10 at the usual time.
- There will be no lecture on Wednesday, Nov 23, the day before
Thanksgiving. The make-up lecture is scheduled for Monday, Nov 21.
- Lecture 1: Complex differentiable functions. Comparison
with real differentiability. Cauchy-Riemann equations. Harmonic
- Lecture 2: Geometric interpretation of
differentiability. Conformal maps. Power series. Radius of
convergence. Analytic functions.
- Lecture 3: Differentiability of power series. Branches of the
logarithm. Principal branch. nth roots. Integration over continuously
differentiable paths. Basic properties and examples.
- Lecture 4: Some results about existence of primitives. Goursat's lemma. Existence of primitives of complex
differentiable functions defined over discs. Integration over continuous
- Lecture 5: Integration over continuous path and
the main results stated.
Winding number. Homotopies. Existence of primitives and
branches of the logarithm in simply connected sets.
- Lecture 6: Integrals of
differentiable functions depend on the homotopy class of paths/loops.
Enhancements. Cauchy's integral formula.
- Lecture 7: Differentiable functions and Taylor series.
Cauchy's integral formula for derivatives. Cauchy's estimates.
Liouville's theorem. Fundamental theorem of algebra. Zeroes of
- Lecture 8: Identity principle. Inverse function theorem. Open
mapping theorem. Maximum modulus principle.
- Lecture 9: Laurent expansion. Types of singularities.
Removable singularity theorem. Characterization of poles. Meromorphic
- Lecture 10: Essential singularities and their
characterization. Computation of residues. Residue theorem stated and the
argument in simple cases.
- Lecture 11: Chains. Enhancements of Cauchy's theorem. Proof of
the residue theorem.
- Lecture 12: Applications of the residue theorem. Integrals of
rational functions. Trigonometric integrals. Examples from Fourier
- Lecture 13: More examples of integrals from Fourier analysis.
Logarithmic integrals. Mellin transform.
- Lecture 14: Residues of forms and residues at infinity. Sums
of residues equals zero. Order of meromorphic functions and the
logarithmic derivative. Argument principle.
- Lecture 15: Applications of the argument principle. Elliptic
functions. Number of zeros and poles. Sum of zeros and poles. Rouche's
- Lecture 16: Sequences and series of holomorphic functions.
Weierstrass convergence theorem. Hurwitz theorem. Limits of injective
functions. Infinite products.
- Lecture 17: Convergence and absolute convergence of infinite
products. Infinite products of holomorphic functions.
Logarithmic derivative of infinite products. Product expansion for the
- Lecture 18: Weierstrass problem. Weierstrass factorization.
Every meromorphic function on C is quotient of entire functions.
Homework 1 due Wednesday, October 5 - PDF
Homework 2 due Friday, October 14 - PDF
Homework 3 due Friday, October 21 - PDF
Homework 4 due Friday, October 28 - PDF
Practice problems for the midterm - PDF
Midterm from Fall 2015 - PDF
Midterm - PDF
Homework 5 due Friday, November 18 - PDF
Homework 6 due Friday, December 3 - PDF
Final from Fall 2015 - PDF. We did not
discuss the material needed for 3 and 8(ii) this quarter.
Topics for Final and Practice Sheet - PDF
Final Exam - PDF