This weekly summary of the lectures is provided to help you make sure you are absorbing the material. Ideally, you should be able to back
up all statements with your own arguments, intuition, examples
Winter 2014
Math 249
Summary Week 2 (January 14&16)
Important Note: This weekly summary of the lectures is provided to help you make sure you
are absorbing the material. Ideally, you should be able to
MATH 249 Assignment 1
Due: Monday, January 25, by 4pm
McGill University
Winter 2016
Please submit your homework to the homework slot at Burnside 1005 on Monday, January 25,
by 4.00pm.
Complex numbers
MATH 249
McGill University
Winter 2016
Practice questions on the evaluation of integrals
Prove the following formulas by using the method of residues:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Z
MATH 249 Assignment 3
McGill University
Winter 2016
Due: Exceptionally Monday, February 22 by 4pm (because of the midterm on the next day).
Solutions will be posted shortly after 4pm. No late assignme
COMPLEX DIFFERENTIABILITY
TSOGTGEREL GANTUMUR
Contents
1.
2.
3.
4.
The problem of extension
Limits and continuity
Complex differentiability
Real differentiability and the Cauchy-Riemann equations
1
4
ISOLATED SINGULARITIES AND MEROMORPHIC FUNCTIONS
TSOGTGEREL GANTUMUR
Contents
1.
2.
3.
4.
5.
6.
Laurent series
Isolated singularities
Residues and indices
The argument principle
Mapping properties of
Winter 2014
Math 249
Summary Week 1 (January 7&9)
Important Note: This weekly summary of the lectures is provided to help you make sure you
are absorbing the material. Ideally, you should be able to b
McGill University
Mathematics and Statistics
Winter 2014 Mathematics 249 (Complex Variables)
Solutions to Midterm Exam
Definitions
(M1) (i) What is a power series?
(ii) What is the radius of convergen
Winter 2014
Math 249
Summary Week 5 (February 4&6)
1
Chapter 1: Preliminaries; 2. Holomorphic Functions
More on power series and analytic functions
We illustrated the issues arising in analytic contin
Summary Week 7 (February 18&20)
(Were skipping the evaluation of real integrals for now; I feel its more pedagogical to treat
them after.)
Chapter 2; Cauchys Theorem; 4. Cauchy Integral formula
Let f
Winter 2014
Math 249
Summary Week 8 (February 25&27)
Chapter 2: Cauchys Theorem; 4. Cauchy Integral Formula
The CIF is the basic tool for connecting holomorphy with analyticity. For example, by
expand
Winter 2014
Math 249
Summary Week 9 (March 11&13)
Applications
Chapter 2: Cauchys Theorem; 5.
Recall (from Week 3) that real and imaginary parts of holomorphic functions are harmonic.
The Cauchy integ
Winter 2014
Math 249
Summary Week 4 (January 28&30)
Important Note: This weekly summary of the lectures is provided to help you make sure you are absorbing the
material. Ideally, you should be able to
Winter 2014
Math 249
Summary Week 3 (January 21&23)
Important Note: This weekly summary of the lectures is provided to help you make sure you are absorbing the material. Ideally, you should be
able to
THE FUNDAMENTAL THEOREMS OF FUNCTION THEORY
TSOGTGEREL GANTUMUR
Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Contour integration
Goursats theorem
Local integrability
Cauchys theorem for homotopic loops