8/4/2019 A. Gioi Thieu Math334

1/2

COMPLEX VARIABLES MAT334

MWF 9am SS 2135

Instructor: Dejan Slepcev

Office: ES 3143 (Earth Sciences building, 22 Russel St.)

Office Hours: Monday 11-1, Wednesday 10-11

Phone: (416) 946-5442

E-mail: slepcev@math.toronto.edu

Course webpage: www.math.toronto.edu/slepcev/M334/

Textbook: Complex Variables by S. Fisher (2nd edition, published by Dover)

Description. This is an introductory course into the analysis of functions of complex variables.

Prerequisite is knowledge of calculus of functions of several variables. The topics covered include:introduction to the complex plane, functions of complex variables and their properties, notion of

analytic functions, Cauchys theorem and integral formula, notion of meromorphic functions and

residue calculus (with application to evaluating integrals), conformal mappings, and introduction to

harmonic functions.

Evaluation. The course grade will be based on homework assignments (10%), two term tests (20%

each), and the final exam (50%). The term tests will be 50 minute tests conducted in class on

February 11 and March 17. Many of the problems will ask you to find something (for example

compute something), but a good number of problems will ask you to show something (in other

words, to prove something). You will not be asked to reproduce proofs of theorems from the

lectures. However you may be asked to give the statements of the theorems.

Homework will be assigned weekly. The problem sets, along with the due dates will be posted on

the course webpage. There will be 11 problem sets (10 best count for the grade). Late homeworks

will not be accepted. Most of the problems will be exercises from the textbook.

APPROXIMATE SCHEDULE

Jan. 5-9 Algebra and geometry of complex numbers (arithmetic, complex conjugation, modulus,

argument, polar representation, roots and powers of complex numbers) . . . . . . . Sec 1.1, 1.2

Jan. 12-16 Topology of the complex plane (interior point, open and closed sets, accumulation

point, boundary of a set, connected sets) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sec. 1.3

Jan. 19-23 Functions of a complex variable (limits, continuity) . . . . . . . . . . . . . . . . . . . . . . Sec. 1.4

Examples of functions (exponential, logarithmic, trigonometric) . . . . . . . . . . . . . . . . . Sec 1.5

Jan. 26-30 Differentiation, Cauchy-Riemann equations, analytic functions Sec 2.1 without 2.1.1

Power series (radius of convergence, differentiation of power series, exponential and trigono-

metric functions as power series) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sec 2.2

Feb. 2-6 Line integrals, Greens theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S ec 1.6

Cauchys Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sec 2.3

Cauchys Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sec 2.3

8/4/2019 A. Gioi Thieu Math334

2/2

Feb. 9-13 Applications of Cauchys formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sec 2.4

First term test, Feb 11Morreras Theorem, Liouvilles Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sec 2.4

Feb. 23-27 Isolated singularities (removable singularities, poles, essential singularities) Sec 2.5

Residues and their computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sec 2.5

Meromorphic functions, Laurent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sec 2.5

Mar. 1-5 Residue theorem and applications to evaluation of definite integrals . . . . . . . . . . Sec 2.6

Mar. 8-12 Zeroes of analytic functions (the argument principle, Rouches theorem, fundamental

theorem of algebra) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sec 3.1 without 3.1.1

Maximum modulus principle and mean-value theorem (Schwartzs lemma) . . . . . . . Sec 3.2

Mar. 15-19 Mobius transformations (that is Linear fractional transformations) . . . . . . . . . Sec 3.3

Second term test, Mar 17Conformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sec 3.4

Mar. 22-26 The Riemann mapping theorem (without a proof) . . . . . . . . . . . . . . . . . . . . . . . . Sec 3.5

Introduction to analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . from lecture notes

Mar. 29-Apr. 2 Harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sec 4.1

P o i s s o n i n t e g r a l f o r m u l a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S e c 4 . 3

Apr. 5-7 Review week