|Office ours:||Wednesday 11-12, Thursday 11-12|
|Time:||Monday, Wednesday, Friday 9:35 AM - 10:25 AM|
|Location:||Burnside Hall 1B39|
|Dates:||Sep 01, 2010 - Dec 03, 2010|
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:
|Hand in on||Exercise||Solutions|
|Wed, Sep 15||Sheet 1||Solutions 1|
|Mon, Sep 20||Sheet 2||Solutions 2|
|Mon, Sep 27||Sheet 3||Solutions 3|
|Mon, Oct 4||Sheet 4|
|Wed, Oct 13||Sheet 5|
|No assignment this week||because of mid-term exam|
|Wed, Nov 3||Sheet 6||Solutions 6|
|Wed, Nov 10||Sheet 7||Solutions 7|
|Fri, Nov 19||Sheet 8||Solutions 8|
|Fri, Nov 26||Sheet 9||Solutions 9|
Exercises are usually to be handed in on Wednesdays.
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, ...
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
Convergence, power series, convergence radius, differentiability
Path integral, fundamental properties of holomorphic functions (Cauchy's and Morera's theorem, etc.), Cauchy integral formula, Lioville's theorem, fundamental theorem of algebra
Elementary domains, connectedness, path connectedness, starlike regions, identity theorem, open mapping theorem, maximum principle, Schwarz lemma, Schwarz-Pick theorem
Uniform and compact convergence, Weierstrass theorem, Laurent series
Singularities, Riemann's theorem on removable singularities, Casorati-Weierstrass theorem, meromorphic functions, winding number, residue theorem, applications, Rouche theorem
Mittag-Leffler theorem, Weierstrass products, product expansion of sine, gamma function
Riemann mapping theorem, the group of conformal mappings
If time is left --- fundamental group, simple connectedness, analytic continuation, coverings
If time is left --- either elliptic functions, modular forms, or Riemann zeta function and prime number theorem
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).