Welcome to Math 220a!

Course description:

This course provides an introduction to complex analysis. We will cover holomorphic and meromorphic functions, Taylor and Laurent expansions, Cauchy's 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, AP&M 6-101.

Lectures: WF, 11:00-12:20, AP&M 5-402.

Office hours:

Wednesday 1-2 in AP&M 6-101.

I am available for questions after lecture or by appointment. Also, feel free to drop in if you see me in my office.

Textbook: Complex Analysis by Lars Ahlfors.

Course Assistant: Pun Wai Tong, p1tong at ucsd dot edu, APM 6-331. Office hour: Tuesday 11-1.

Prerequisites:

The minimal prerequesites are Math 140 A-B. However, this is a graduate level course, so at times, we may use notions from related fields, 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.

Grading:

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.

Important dates:

• 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.

Announcements:

• 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.

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Lecture Summaries

• 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 roots. 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 continuous paths.
• 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 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 holomorphic functions.
• 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 domains.
• 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:

Homework 1 due Wednesday, October 7 - PDF

• Solutions: PDF

Homework 2 due Wednesday, October 14 - PDF

• Solutions: PDF

Homework 3 due Wednesday, October 21 - PDF

• Solutions: PDF

Practice problems for the midterm - PDF

Midterm - PDF

• Solutions: PDF

Homework 4 due Friday, November 13 - PDF

• Solutions: PDF

Homework 5 due Monday, November 23 - PDF

• Solutions: PDF

Homework 6 due Wednesday, December 2 - PDF

• Solutions: PDF

Practice problems for the final - PDF.

Final - PDF

• Solutions: PDF