Textbook: Introduction to Complex Analysis, H. A. Priestley, 2nd ed, Oxford

Course objective: To learn the fundamental ideas in the study of functions of a complex variable. These include: differentiation, integration, power series, residue theorem and the maximum modulus principle.

Topics (chapter references are to the above textbook):

1) algebra, geometry and toplogy in the complex plane -- chapters 1, 2, 3,

2) complex functions: differentiation, power series, examples -- chapters 5, 6, 7

3) integration along paths; Cauchy's theorem -- chapters 4, 10, 11

4) Cauchy's integral formula -- chapter 13

5) power series: zeros and singularities -- chapters 14, 15, 17

6) Cauchy's residue theorem: examples, applications -- chapters 18, 19, 20

7) maximum modulus theorem -- chapter 16

Instructor: Dr. D.
Haskell, HH316, ext.27244

Course meeting time: TRF 8:30 - 9:20 ABB 136

E-mail: haskell@math.mcmaster.ca

Office hours: T 9:30-11:30, R 9:30-10:30

TA Xiaoye Fu, office
hours M 15:00-17:00 behind the Math Cafe

Tutorial: R 10:30-11:20 BSB 136

Course requirements,
in brief (consult the course
information sheet for more detailed information).

Homework: 20%

Midterm I: 20%

Midterm II: 20%

Final: 40%

Announcements

10
April 2011: Solutions to Midterm 2 are posted on the calendar.

8
April 2011: Solutions to Homework 6 are posted on the calendar below.
The marked homeworks are available for pickup in my office.

7
April 2011: I will hold my usual office hours during the exam period
(with two hours on Thursdays). As usual, I am in the office most of the
time, and available to answer questions on a drop-in basis. Some
questions to practice on to prepare for the final are posted here.
Apologies for the poor quality of the copy.

29
March 2011: Homework 6 is posted on the calendar.

24
March 2011: Solutions to Homework 5 posted on the calendar.

15
March 2011: Homework 5 is posted on the calendar below.

15
March 2011: Mathematics is Connected! Students who are also taking Math
3T03 or Math 3EE3 can instead do one or both of Problem 3 (3X/3T) and
Problem 4 (3X/3EE) of the Joint
Homework (which may be handed in to any one of the three
instructors) as a substitute for any two of the problems on Homework
5. Please indicate on your Homework 5 that you have also handed
in the Joint Homework.

9
March 2011: Mostafa has the following correction to 14.1: the taylor
series for sin^2(z) starts with n=1 and has coefficient

(-1)^{n-1}2^{2n-1}/(2n)! for z^{2n}. Also note that it is much easier
to find the taylor series using trigonometric identities

than by either multiplying the series for sine by itself, or by
differentiating sin^2.

8
March 2011: Solutions to Homework 4 are posted on the calendar.

7
March 2011: Thanks to Filip for the following corrections to the
solutions posted below.

4.2
(ii) The semi-circle is supposed to be closed

10.1 (iv) The parametrization for [1,1 + i] is incorrect

10.1 (v) The integral you have there evaluates to 56/15

10.2 (ii) cos(e^it)^2 should be (cos(t))^2 [and hence the integral
needs to be done using integration by parts]

3
March 2011: Midterm 2 is on Thursday, March 3, in class. It will cover
chapters 10, 11, 13 and 14 from the textbook, as well as the
small amount of ch. 4 that we discussed in class. Good problems to work
through to prepare for the midterm are the following:

4.2 solutions

10.1, 10.2 solutions

11.1, 11.2, 11.3 solutions

13.1, 13.9, 13.11 solutions

14.1 solutions

Solutions
will be posted soon.

3
March 2011: Solutions to Homework 3 are posted on the calendar.

16
February 2011: Homework 4 is posted on the calendar.

11
February 2011: Note that in Question 3(i) of Homework 3, the curve is
not closed.

10
February 2011: Solutions to Midterm 1 are posted on the calendar.

8
February 2011: Midterm 1 was handed back in class today. Please look
over your exam. If you think there is a mistake in the grading, bring
the exam to me to review. Several mistakes have been found already,
resulting in significant changes in marks, so please check.

3
February 2011: Homework 3 is posted below. Homework 2 and the midterm
will be handed back on Tuesday, 8 February.

1
February 2011: Solutions to Homework 2 posted on the calendar.

28
January 2011: Midterm 1 coming up next week (Thursday). It will cover
chapter 1, 2, 3, 5, 6, 7 of the textbook. Use the recommended problems
for review.

21
Jan 2011; Solutions to Homework 1 posted on the calendar.

13
Jan 2011: Homework 1 Question 3 (problem 2.14 of Priestley) has two
typos. In (ii)(b), the equation of the arc is arg((z-alpha)/(z-beta)) =
mu (so division, rather than multiplication). Also in (iii) K has a c
in the numerator rather than a 1.

Also, in Question 5, you do not need to prove
rigorously that the sets are connected. Note that the definition I gave
is that for an open set to be connected. If G is not open, still it is
connected if it cannot be written as the disjoint union of two sets
which are open as subsets of G.
If you don't know what this means, work the problem at a more intuitive
level.

Course Calendar

Dates |
Tuesday | Thursday |
Friday | Recommended Problems |

Week 1 Jan 3 - 7 |
algebra, geometry and
topology in the complex plane |
algebra, geometry and topology in the complex plane | class cancelled today |
Priestley Ch 1 pp 9-11: 1.1,
1.2, 1.3, 1.6, 1.7, 1.10 |

Week 2 Jan 10 - 14 |
algebra, geometry and topology in the complex plane | algebra, geometry and topology in the complex plane | algebra, geometry and topology in the complex plane | Priestley Ch 2 pp 26-29:
2.1--2.6, 2.11 Priestley Ch 3 pp 44--46: 3.1--3.4, 3.5, 3.7, 3.13 |

Week 3 Jan 17 - 21 |
complex functions: differentiation, power series, examples Homework 1 due in class solutions |
complex functions: differentiation, power series, examples | complex functions: differentiation, power series, examples | Priestley Ch 5 pp 64-66: 5.2,
5.4, 5.7, 5.9, 5.11 Priestley Ch 6 pp 76-77: 6.2, 6.3 |

Week 4 Jan 24 - 28 |
complex functions: differentiation, power series, examples | complex functions: differentiation, power series, examples | complex functions: differentiation, power series, examples | Priestley Ch 6 pp 76-77: 6.4, 6.7 Priestley Ch 7 pp 89-90: 7.1, 7.2, 7.4, 7.7, 7.8, 7.9, 7.11, 7.14 |

Week 5 Jan 31 - Feb 4 |
integration along paths; Cauchy's theorem Homework 2 due in class solutions |
Midterm 1 solutions |
integration along paths; Cauchy's theorem | Priestley Ch 4 p 55: 4.1, 4.2 Priestley Ch 10 pp 126-127: 10.1, 10.2, 10.5, 10.6 |

Week 6 Feb 7 - 11 |
integration along paths; Cauchy's theorem | integration along paths; Cauchy's theorem | Cauchy's integral formula | Priestley Ch 11 pp 140-141:
11.1, 11.2, 11.3, 11.7 |

Week 7 Feb 14 - 18 |
Cauchy's integral formula Homework 3 due in class solutions |
Cauchy's integral formula | power series: zeros and singularities | Priestley Ch 13 pp 159-160:
13.1, 13.2, 13.4, 13.5, 13.8, 13.9, 13.11 |

Feb 21 - 25 |
READING WEEK |
READING WEEK | READING WEEK | |

Week 8 Feb 28 - Mar 4 |
power series: zeros and singularities | power series: zeros and singularities | power series: zeros and singularities |
Priestley Ch 14 pp 174-175: 14.1, 14.2, 14.4, 14.5, 14.6 |

Week 9 Mar 7 - 11 |
power series: zeros and singularities Homework 4 due in class solutions |
Midterm 2 solutions |
power series: zeros and singularities | Priestley Ch 15 pp 185-187:
15.1, 15.3, 15.4, 15.9, 15.10, 15.11, 15.13, 15.14 Priestley Ch 17 pp 207-210: 17.1, 17.5, 17.6, 17.8 |

Week 10 Mar 14 - 18 |
power series: zeros and singularities | power series: zeros and singularities | Cauchy's residue theorem: examples,
applications |
Priestley Ch 17 pp 207-210:
17.9, 17.10, 17.12, 17.19(i)-(iv) Priestley Ch 18 pp 219-220: 18.2, 18.3, 18.4 |

Week 11 Mar 21 - 25 |
Cauchy's residue theorem: examples, applications Homework 5 due in class solutions |
Cauchy's residue theorem: examples, applications | maximum modulus theorem | Priestley Ch 18 pp 219-220:
18.8, 18.9 Priestley Ch 20 pp 251--255: 20.1, 20.2, 20.3, 20.4, 20.13 |

Week 12 Mar 28 - Apr 1 |
Cauchy's residue theorem: examples, applications | Cauchy's residue theorem: examples, applications | Cauchy's residue theorem: examples, applications | |

Week 13 Apr 4 - 8 |
review Homework 6 due in class solutions |