Course objective: To learn the fundamental and beautiful 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 -- sections 1.1, 1.2

2) complex functions: examples, continuity, differentiability -- sections 1.3, 1.4, 1.5, 1.6

3) contour integrals and Cauchy's theorem -- sections 2.1, 2.2

4) Cauchy's integral formula and the maximum modulus theorem -- sections 2.4, 2.5

5) power series and Laurent series --- 3.1, 3.2, 3.3

6) calculus of residues --- 4.1, 4.2, 4.3

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

Course meeting time: MR 9:30 - 10:20, T 10:30 - 11:20 BSB 136

E-mail: haskell@math.mcmaster.ca

Office hours: M 10:30--11:30, W 10:30--12:30

TA Jizhan Hong
hongj37@math.mcmaster.ca

Tutorial: T 11:30-12:20 BSB 136

Office hours: TW 4:30--5:30 (pm) behind the Math Cafe

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

Homework: 20%

Midterm I: 15%

Midterm II: 15%

Final: 50%

Announcements

11 April 2012: Preparatory remarks for the final exam: You do not need to know precise statements of theorem. The exam will not ask for any standard proofs. You should be able to use theorems including: funadamental theorem for contour integrals, Cauchy's theorem, path independence theorem, maximum modulus principle, Schwarz Lemma, Taylor's theorem and Laurent series, residue theorem. "Use" means that you can check that the theorem applies by verifying the hypotheses, and draw a conclusion.

Here are some good problems to practice on:

Chapter 4 Review: 3, 7, 13, 15, 17, 21, 23

Chapter 3 Review: 5, 13, 25

Chapter 2 Review: 1, 5, 21

Chapter 1 Review: 1, 11, 15

11 April 2012: Office hours before the final exam:

Jizhan --

Apr. 18 (Wed): 2pm - 3pm

Apr. 19 (Thu): 10am- 12pm

Apr. 20 (Fri): 10am- 12pm

DH --

Wednesday, April 18, 10-12

Friday, April 20, 1-3

or by appointment

11 April 2012: Here are all the marks for the term, sorted by last five digits of student id number. Please check that I have everything recorded correctly. As always, a blank indicates no mark recorded, and an X indicates an excused absence. As promised in class, I will compute the final mark in two ways: one as originally announced and one with 75% on the final exam. You will recieve the better of the two marks.

18 March 2012: Homework 6 is posted.

16 March 2012: Solutions to homework 5.

16 March 2012: Joke of the week

Q. What is the contour integral around Western Europe?

A. Zero, because all the Poles are in Eastern Europe.

Addendum: there are poles in Western Europe, but they are all removable.

1 March 2012: Solutions to homework 4. Also, homework 5 is posted below.

29 February 2012: Complete solution to a problem that Jizhan was explaining in the tutorial.

27 February 2012: Midterm 2 will be in T29 105 on Monday, 5 March 9:30-10:20. You should know the hypotheses and conclusions of the following theorems:

2.2.3 Cauchy's theorem

2.2.4 path independence theorem

2.4.4/2.4.6 Cauchy's integral formula and for deriviatives

2.4.8 Liouville's theorem

2.4.9 Fundamental theorem of algebra

2.5.1/2.5.6 local and global version of the maximum modulus principle

You should know the proofs of the following theorems:

2.4.4 Cauchy's integral formula

2.4.8 Liouville's theorem (as done in class, so cannot just quote the "Cauchy inequalitites")

2.4.9 Fundamental theorem of algebra

There will also be some calculations and applications of theorems.

24 February 2012: Midterm 2 is coming up in a week's time. It will cover all of chapter 2. The review problems in that chapter (except the ones involving harmonic functions) are a good review for the midterm.

12 February 2012: Joke of the week Q. Why did the mathematician name her dog Cauchy?

A. Because he left a residue at every pole.

(Actually, we will understand this joke better in a couple of weeks.)

12 February 2012: Solutions to Homework 3.

27 January 2012: Solutions to Homework 2.

25 January 2012: Midterm 1 wil be held in T29 105 on Thursday 2 February, 9:30-10:30.

24 January 2012: CHANGE in midterm date. Because of the risk of a bus strike, the midterm is moved to THURSDAY 2 February. The room will be posted. Class on Monday is cancelled. There will be class and tutorial on Tuesday.

23 January 2012: Midterm 1 is on Monday, 30 January. It will be held in T29-105. It covers all of chapter 1. The review problems in that chapter are a good review for the midterm.

23 January 2012: Joke of the week Life is complex. It has real and imaginary parts. And the rational parts have measure zero compared to the irrational parts.

18 January 2012: Solutions to Homework 1 are available, thanks to Jizhan.

18 January 2012: Joke of the week \pi to i: "Get real!" i to \pi: "Get rational!"

11 January 2012: Joke of the week In a written exam in freshman calculus, a student solves the equation sin x=2. He writes: x=arcsin(2), and gets an "F". He comes to ask what was wrong, and his professor explains that arcsin(2) does not exist, and that the equation has no solutions. A few years later the same student has an exam in complex analysis with the same professor. He is very glad to see at least one problem, whose solution he knows: to solve the equation sin z=2...

6 January 2012: Amendments to Homework 1: 1.3.3 should be 1.3.4, and 1.4.14 is postponed to the next homework.

3 January 2012 Joke of the week The number you have dialed is imaginary. Please rotate your phone 90 degrees and try again.

9 December 2011 Classes begin Tuesday, 3 January 2012. Until then, have a good holiday.

Course
Calendar

Homework 1, due Thursday 12 January 2012 see amendments above

1.1.10, 1.1.18,

1.2.8, 1.2.24,

1.3.3, 1.3.10, 1.3.14, 1.3.22

1.4.14

Homework 2, due Thursday 26 January 2012

1.4.2, 1.4.8, 1.4.14, 1.4.16

1.5.2, 1.5.4, 1.5.6, 1.5.10,

1.6.2, 1.6.4

Challenge problem (not required): 1.5.16

Homework 3, due Thursday 9 February 2012

2.1: 2, 8, 10

2.2: 2, 6, 8

2.3: 8

Homework 4, due Thursday 1 March 2012

2.4: 2, 4, 5, 6, 8, 16

2.5: 2, 5, 7

Homework 5, due Thursday 15 March 2012

3.1: 7, 10, 14

3.2: 2, 6, 8, 18

Extra credit problem: 3.2.20

Homework 6, due Thursday 29 March 2012

3.3: 2, 4, 10

4.1: 2, 8

4.2: 2, 10

Dates |
Monday | Tuesday |
Thursday | Recommended Problems |

Week 1 Jan 3 - 6 |
no class |
complex numbers - algebra and geometry 1.1, 1.2 |
complex numbers - algebra and geometry 1.1, 1.2 | 1.1: 1, 5, 7, 17 1.2: 1, 3, 9, 19, 23 |

Week 2 Jan 9 - 13 |
examples of complex functions 1.3 |
analysis in the complex plane
1.4 |
analysis in the complex plane
1.4 Homework 1 due |
1.3: 1, 3, 9, 13, 23 1.4: 5, 7, 15 |

Week 3 Jan 16 - 20 |
continuity and differentiability of complex functions 1.5 |
continuity and differentiability of complex functions 1.5 |
continuity and
differentiability of complex functions 1.5 |
1.5: 1, 5, 9, 25 |

Week 4 Jan 23 - 27 |
some specific derivatives 1.6 |
contour integrals 2.1 |
contour integrals 2.1 Homework 2 due |
1.6: 1, 5, 9 2.1: 1, 7, 9, 11 |

Week 5 Jan 30 - Feb 3 |
Midterm 1 no class today |
Cauchy's theorem 2.2 |
Midterm 1 | 2.2 1, 3, 9 |

Week 6 Feb 6 - 10 |
Cauchy's integral formula 2.4 | Cauchy's integral formula 2.4 |
Homework 3 due | 2.4: 1, 3, 13, 15, 17 |

Week 7 Feb 13 - 17 |
Cauchy's integral formula 2.4 | maximum modulus 2.5 | maximum modulus 2.5 |
2.5: 1, 3 |

Feb 20 - 24 |
Reading Week |
Reading Week | Reading Week | |

Week 8 Feb 27 - Mar 2 |
harmonic
functions Schwarz lemma |
convergent series 3.1 |
convergent series 3.1 Homework 4 due |
3.1: 2, 9, 12 |

Week 9 Mar 5 - 9 |
Midterm 2 |
power series 3.2 | power series 3.2 |
3.2: 1, 3, 7, 9 |

Week 10 Mar 12 - 16 |
Laurent series 3.3 | Laurent series 3.3 |
calculation of residues 4.1 Homework 5 due |
3.3: 1, 3 4.1: 1, 5 |

Week 11 Mar 19 - 23 |
calculation of residues 4.1 |
residue theorem 4.2 | residue theorem 4.2 |
4.2: 1, 3, 9 |

Week 12 Mar 26 - 30 |
evaluation of integrals 4.3 | evaluation of integrals 4.3 |
evaluation
of integrals 4.3 Homework 6 due |
4.3: 1, 3, 5, 7 |

Week 13 Apr 2 - 4 |
evaluation of integrals 4.3 | concluding remarks |