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. 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, AP&M 6-101.

Lectures: WF, 12:30-1:50, AP&M B-412.

Office hours:

Wednesday 2-4 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.

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 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.

Important dates:

• 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.

Announcements:

• 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.

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Lecture Summaries

• Lecture 1: Complex differentiable functions. Comparison with real differentiability. Cauchy-Riemann equations. Harmonic functions.
• 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 paths.
• 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 holomorphic functions.
• 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 functions.
• 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 analysis.
• 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 theorem. Examples.
• 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 sine function.
• Lecture 18: Weierstrass problem. Weierstrass factorization. Every meromorphic function on C is quotient of entire functions. Genus. Examples.

Homework:

Homework 1 due Wednesday, October 5 - PDF

• Solutions: PDF

Homework 2 due Friday, October 14 - PDF

• Solutions: PDF

Homework 3 due Friday, October 21 - PDF

• Solutions: PDF

Homework 4 due Friday, October 28 - PDF

• Solutions: PDF

Practice problems for the midterm - PDF

Midterm from Fall 2015 - PDF

• Solutions: PDF

Midterm - PDF

• Solutions: PDF

Homework 5 due Friday, November 18 - PDF

• Solutions: PDF

Homework 6 due Friday, December 3 - PDF

• Solutions: PDF

Final from Fall 2015 - PDF. We did not discuss the material needed for 3 and 8(ii) this quarter.

• Solutions: PDF

Topics for Final and Practice Sheet - PDF

Final Exam - PDF

• Solutions: PDF