MATH 316: Complex Analysis
Fall 2010
Course description

Instructor

Course information

There will be a mid-term and final exam, as well as weekly exercises. The final grading will be based on the exercises (20 %), mid-term exam (20 %) and final exam (60 %).

Solutions to the midterm exam:
midterm exam

Exercises

Exercises are usually to be handed in on Wednesdays.

Textbooks

Recommended:

1. Ash, Robert B.; Complex variables.Academic Press, New York-London 1971, viii+255 pp.
   New version available at: http://www.math.uiuc.edu/~r-ash/CV.html

Other:

1. Freitag, Eberhard; Busam, Rolf; Complex analysis. Second edition. Universitext. Springer-Verlag, Berlin, 2009. x+532 pp. ISBN: 978-3-540-93982-5
   Available as ebook: http://mcgill.worldcat.org/title/complex-analysis/oclc/288311074
2. Conway, John B.; Functions of one complex variable. Second edition. Graduate Texts in Mathematics, 11. Springer-Verlag, New York-Berlin, 1978. xiii+317 pp. ISBN: 0-387-90328-3
3. Lang, Serge; Complex analysis. Fourth edition. Graduate Texts in Mathematics, 103. Springer-Verlag, New York, 1999. xiv+485 pp. ISBN: 0-387-98592-1
4. Wunsch, A. David; Complex variables with applications. Second edition. Addison-Wesley Publishing Company, Reading, MA, 1994. xii+622 pp. ISBN: 0-201-12299-5

Syllabus

Overview and complex numbers

Motivation and definition of complex numbers, comparison of real and complex analysis, overview of the course, some properties of complex numbers: complex conjugation, norm, polar coordinates, roots, ...

Holomorphic functions

Möbius transformations, Riemann sphere, half plane and unit disc, topology of C, continuity, complex differentiability, holomorphy, Cauchy-Riemann differential equations

Download a visualization of Möbius transformations: download

A nice animation: Möbius Transformations Revealed

Sequences and series

Convergence, power series, convergence radius, differentiability

Complex integration

Path integral, fundamental properties of holomorphic functions (Cauchy's and Morera's theorem, etc.), Cauchy integral formula, Lioville's theorem, fundamental theorem of algebra

Domains

Elementary domains, connectedness, path connectedness, starlike regions, identity theorem, open mapping theorem, maximum principle, Schwarz lemma, Schwarz-Pick theorem

Families of holomorphic functions

Uniform and compact convergence, Weierstrass theorem, Laurent series

Singularities and Residues

Singularities, Riemann's theorem on removable singularities, Casorati-Weierstrass theorem, meromorphic functions, winding number, residue theorem, applications, Rouche theorem

Construction of meromorphic functions

Mittag-Leffler theorem, Weierstrass products, product expansion of sine, gamma function

Conformal mappings

Riemann mapping theorem, the group of conformal mappings

Domains II

If time is left --- fundamental group, simple connectedness, analytic continuation, coverings

Applications

If time is left --- either elliptic functions, modular forms, or Riemann zeta function and prime number theorem

Notice

McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offenses under the Code of Student Conduct and Disciplinary Procedures (see McGill web page on Academic Integrity for more information).