# Math 566

**Catalog description**

Simple connectivity, use of logarithms;
argument, conservation of domain and maximum principles;
analytic continuation, monodromy theorem;
conformal mapping; normal families, Riemann mapping theorem;
harmonic functions, Dirichlet problem;
introduction to functions of several complex variables.

**Topics to be covered**

We plan to cover at least a good part of the following topics.

*Harmonic functions*: the Laplace equation, the mean value property, regularity, the Schwarz reflection principle, the maximum principle, uniform limits, the Harnack principle, the Dirichlet problem, subharmonic functions, Perron's method, Green's function, Poisson's formula for a disc

*Holomorphic functions*: the Cauchy-Riemann equation, Cauchy's theorem for simple contours, Goursat's theorem, the Cauchy integral formula, Pompeiu's formula, Morera's theorem, Liouville's theorem, the Schwarz reflection principle, the roots and logarithm functions, multi-valued functions, Riemann surface, branch points, covering spaces, universal cover, maximum modulus principle, Hadamard three-circle theorem, Hardy's theorem, Borel-Carathéodory theorem, Phragmén-Lindelöf principle, Carlson's theorem, Kramers-Kronig relations

*Analyticity*: power series, the identity theorem, uniqueness principle, the principle of permanence of functional equations, analytic continuation, the monodromy theorem, the open mapping theorem, the Laurent decomposition, the Laurent series, Riemann's removable singularity theorem, Casorati-Weierstrass theorem, periodic functions

*Meromorphic functions*: the residue theorem and its application to computation of integrals, the argument principle, Rouché's theorem, Hurwitz's theorem, winding number, homology, homotopy, the jump theorem, Cauchy's theorem for simply connected domains, characterizations of simple connectivity

*Conformal mappings*: elementary mappings, Schwarz's lemma, hyperbolic geometry, the Riemann sphere, Pick's lemma, Zalcman's lemma, Marty's theorem, Montel's theorem, the Picard theorems, the Riemann mapping theorem, the Schwarz-Christoffel formula, Carathéodory's theorem

*Approximation theory*: uniform approximation, Runge's theorem, Jensen's formula, the Mittag-Leffler theorem, Weierstrass' theorems, Mergelyan's theorem, the Hadamard factorization theorem

Application to number theory: Γ and ζ functions, arithmetic functions, Dirichlet series, the prime number theorem

*As time permits*: introduction to several complex variables, introduction to complex dynamics, introduction to elliptic functions, introduction to geometric function theory, Paley-Wiener theorems, introduction to Hardy spaces

**Prerequisites**

MATH 366 (formerly MATH 466), MATH 564. Informally, upper-undergraduate level real analysis and some exposure to elementary complex variables would be sufficient.

**References**

Reinhold Remmert, *Theory of complex functions*. Springer 1991.
Reinhold Remmert, *Classical topics in complex function theory*. Springer 2010.
Theodore W. Gamelin, *Complex Analysis*. Springer 2001.
Raghavan Narasimhan and Yves Nievergelt, *Complex analysis in one variable*. Birkhäuser 2001.
John B. Conway, *Functions of one complex variable I, II*. Springer 1978.
Elias M. Stein and Rami Shakarachi, *Complex Analysis*. Princeton 2003.
Kunihiko Kodaira, *Complex analysis*. Cambridge 2007.
Eberhard Freitag and Rolf Busam, *Complex Analysis*. Springer 2005.
Robert E. Greene and Steven G. Krantz, *Function theory of one complex variable*. AMS 2006.
Steven G. Krantz, *Geometric function theory*. Birkhäuser 2006.
William A. Veech, *A second course in complex analysis*. Dover 2008.
Harold M. Edwards, *Riemann's zeta function*. Dover 2001.
Alexander Ivić, *The Riemann zeta-function: Theory and applications*. Dover 2003.
Samuel J. Patterson, *An introduction to the theory of the Riemann zeta-function*. Cambridge 1995.

**Lectures**

**TuTh 10:05am–11:25am**,
Burnside Hall 1214

**Instructor**

Dr. Gantumur Tsogtgerel

Office: Burnside Hall 1123. Phone: (514) 398-2510. Email: gantumur -at- math.mcgill.ca.

Office hours: Just drop by to check if I am in or make an appointment

**Grading**

The final course grade will be the weighted average of homework 40%, the take-home midterm exam 20%, and the final project 40%.

**Homework**

Assigned and graded roughly every week except a few weeks during the midterm exam and the final project.

**Midterm exam**

The midterm will be a take-home exam.

**Final project**

The final project consists of the student reading a paper or monograph on an advanced topic, typing up notes, and possibly presenting it in class.
A list of topics for the final project will be given out after the midterm exam.