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, and counter-examples. It is not meant as a substitute
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 back up all statements with your own
arguments, intuit
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
Exercise 1. [10 points] Find the real and imaginary par
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
Z
Z
Z
Z
1
0
1
x4
0
dx
= p
2
+x +1
2 3
1
(x2
0
2
0
dx
2
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 assignments will be accepted.
Power series
Exercise 1. [22 poin
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
6
10
1. The problem of extension
In complex analysis, w
ELEMENTARY FUNCTIONS
TSOGTGEREL GANTUMUR
Contents
1.
2.
3.
4.
5.
6.
Constant functions
The exponential
The argument
Logarithms
Powers
Circular functions
1
2
6
8
11
12
1. Constant functions
Before delving into the study of elementary functions, we prove he
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 holomorphic functions
The Riemann sphere and meromorphi
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 back up all statements with your
own arguments, intuitio
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 convergence of a power series?
(iii) Give one way to calculate t
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 continuation: It is singularities at the circle of
convergenc
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 : C be a holomorphic function. For any z, z0 , R > 0 wi
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
expanding the integrand in a geometric series, we learn that
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 integral formula can be used to solve the Dirichlet problemF
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 back up all statements with your own arguments, intuit
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 back up all statements with your own arguments, intuit
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
Evaluation of real definite integrals
The Cauchy integ