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
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 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
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 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 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
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 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