# Home

## What is Complex Analysis?

Complex Analysis is the study of complex-valued functions of a complex variable. As in an introductory calculus course, the first order of business is, of course, defining and exploring the appropriate number system — real numbers then, complex numbers now. After doing that we will define what these functions are, give the most important examples of them, and elucidate what it means to differentiate and integrate them.

## About the course (and this website)

This website contains (hopefully) all of the information you need about the course — except for the content itself, for which you will need to come to class. Please explore all sections. If you have any questions, contact me or your TA.

## Important dates

As discussed in the Other rules section, announcements will be sent through Portal. The important dates, however, are the homework due dates — specified in the Homework section of this webpage — and the exam dates, which are as follows:

• Midterm 1: Oct 6, 4–6pm, EX 320 — Solutions
• Midterm 2: Nov 10, 4–6pm, EX 300 — Solutions
• Final Exam: Dec 19, 6:30–9:30pm, BA 2145

# Staff

## Course instructor

Name:
Alberto García-Raboso
Office:
BA 6172
Email:
agraboso@math.toronto.edu
Office hours:
By appointment only.
Lectures:
Wed 11am-noon — WB 119
Thu 4-6pm — BA 1240

## Teaching Assistant for Section 1

Name:
Evan Miller
Office:
BA 6191
Email:
epmiller@math.utoronto.ca
Office hours:
Thu 12:30-1:30pm — BA 6283
Tutorial session:
Thu 11-12am — WB 144

## Teaching Assistant for Section 2

Name:
Nikita Nikolaev
Office:
BA 6191
Email:
nikolaev@math.utoronto.ca
Office hours:
Tue 7-8pm — WB 144
Tutorial session:
Tue 6-7pm — WB 144

# References

There is no official textbook for the course: most introductory texts cover essentially the same topics, and you are welcome to consult any one of them. Here is a selection.

• [CBx] R.V. Churchill, J.W. Brown, Complex Variables and Applications, McGraw-Hill, 1948-2008.
An absolute classic, and the one from which I learned Complex Analysis. Newer editions are obscenely expensive. If you come across an older one at a decent price, you might want to pick it up.
• [BMPS] M. Beck, G. Marchesi, D. Pixton, L. Sabalka, A First Course in Complex Analysis, available online.
Good lecture notes. And the price is unbeatable!
• [F] S.D. Fisher, Complex Variables, 2nd edition, Dover Publications, 1999.
This was the textbook for this course a few years ago. It is published by Dover, so it is very fairly priced.
• [P] H.A. Priestley, Introduction to Complex Analysis, 2nd edition, Oxford University Press, 2003.
A little gem. Note that it has undergone a significant change from the first edition.
• [A] L.V. Ahlfors, Complex Analysis, McGraw-Hill, 1953-1980.
Another classic, but a level up from what we are aiming for. Once again, the newer editions are very expensive (it is from the same publisher as Churchill and Brown’s book…).
• [K] Y.K. Kwok, Applied Complex Variables, 2nd edition, Cambridge University Press, 2010.
This book intersperses theory and applications.
• [SS] E.B. Saff, A.D. Snider, Fundamentals of Complex Analysis, Pearson Education, 1976-2003.
A quite well-known book that some people love. I have never used it though, so I will withhold judgment.

I will mainly follow the first reference, but not to the letter — I may jump around, cover things that are not there, skip some parts, etc.

# Syllabus

1. Complex Numbers
• [CB5] 1
• [CB8] 1
• [BMPS] 1
• [P] 1, 2, 3
2. Analytic Functions
• [CB5] 2, 8
• [CB8] 2, 9
• [BMPS] 2, 6
• [P] 5, 8, 9, 16, 23
3. Elementary Functions
• [CB5] 3, 7
• [CB8] 3, 8
• [BMPS] 3
• [P] 7
4. Integrals
• [CB5] 4
• [CB8] 4
• [BMPS] 4, 5
• [P] 4, 10, 11, 12, 13, 15, 19, 20
5. Series
• [CB5] 5
• [CB8] 5
• [BMPS] 7, 8
• [P] 6, 14
6. Residues and Poles
• [CB5] 6
• [CB8] 6, 7
• [BMPS] 9
• [P] 17, 18

# Marking

## General scheme

Your final mark in the course will be calculated as follows:

• Homework: 15%
• Quizzes: 15%
• Exams: 70%

## Homework

Weekly problem sets will be posted on the Homework section of this webpage. From those, you will have to turn in a selection of problems through the online platform Crowdmark. For each assignment you will receive an email with a personalized URL where you can make your submission.

These assignments will be due Fridays at 5pm. Absolutely no late homework will be accepted. In order to accomodate eventualities, your lowest two scores will not be counted towards your 15% of the final mark in the course.

You can work with your classmates in solving the problems, but the write-up should be exclusively yours: do not blatanly copy your classmates’ work. Students caught with identical submissions will be given a mark of zero and referred to the appropriate Tribunal.

Of course, a minimum standard of cleanliness and readability is expected of your work (think of your TA!): illegible submissions will mercilessly be given a mark of zero.

The rest of the problems in the sets do not count towards your mark in the course. However, I strongly encourage you to work on as many of them as possible. Remember: mathematics is not a spectator sport!

Solutions will be posted on this webpage a few days after the submission deadline for each assignment.

## Quizzes

There will be a weekly quiz — to be held at the end of each tutorial session — based on the material covered in the homework set due the previous week. It will consist of one problem, for which you will be given ten minutes.

Once again, your two lowest marks will not be counted towards your final mark in the course.

The same standards of cleanliness, readability and academic honesty apply to quizzes.

## Exams

Under normal circumstances, the 70% corresponding to exams will be broken down as follows:

• Midterm 1: 15%
• Midterm 2: 20%
• Final Exam: 35%

Should you miss one midterm due to illness (and provide me with appropriate documentation), the above weights will be modified to:

• Midterm: 25%
• Final Exam: 45%

Missing both midterms (again, with supporting documentation) will shift your whole exam mark to your performance on the final exam:

• Final Exam: 70%

And if you cheat in any exam, there is a special kind of hell reserved for you…

# Other rules

• Class announcements and Crowdmark assignments will be sent through Portal. You are responsible for making sure that you are registered on the class email list for Portal and for checking that email regularly. If you normally use another email address then you must make sure you check your U of T email regularly or have it forwarded to your usual address.
• Marks will also be posted on Portal.
• If you need to communicate with me, please do so right before or after class, or through email. Do not drop by my office unannounced. You should pose content-related questions in person, either before or after class or during office hours. Email is a poor medium for this kind of interaction; as such, I reserve the right to not answer any emails of a mathematical nature.
• As already remarked in the Marking section, you must always abide by the University’s Code of Behaviour on Academic Matters.
• If one or both of the midterms happen to fall on a religious holiday that you observe, let me know as soon as possible. If a sizable number of students is affected, we might need to reschedule them.
• Students with diverse learning styles and needs are welcome in this course. In particular, if you have a disability/health consideration that may require accommodations, please feel free to approach me and/or Accessibility Services at (416) 978-8060 or online.

# Lectures

Date Topics covered
Sep 8 Chapter 1
• Definition of complex numbers
• Polar form: modulus and argument
• Euler's formula
• Complex conjugation
• The triangle inequality
Sep 14 • Powers, roots and roots of unity
• Functions as transformations
• Translations, dilations and rotations
Sep 15 • Affine linear transformations
• Inversion
• The extended complex plane
• Stereographic projection and the Riemann sphere
Sep 21 • Möbius transformations
Sep 22 • More Möbius transformations
• Topology of the complex plane
Chapter 2
• Functions and their domains
• Examples of functions
Sep 28 • Limits
• Continuity
• Limits involving the point at infinity
Sep 29 • Definition of complex derivative
• Differentiation rules
• The Cauchy-Riemann equations
• Sufficient conditions for differentiability
• Definition of holomorphic functions
Oct 5 • Definition of harmonic functions
• Harmonic functions from holomorphic functions
• Harmonic conjugates
Oct 6 NO CLASS: Midterm 1
Oct 12 • Existence of harmonic conjugates
Oct 13 • Conformality
• Critical points of holomorphic functions
• The inverse function theorem
Chapter 3
• The exponential function
Oct 19 • Trigonometric and hyperbolic functions
Oct 20 • Logarithms
• Combining branch cuts
Oct 26 • Complex exponents
Oct 27 • Exponentials with other bases
• Inverse trigonometric functions
Chapter 4
• Reminder on the Riemann integral
• Integrals of complex-valued functions on the real line
• Complex integrals: definition and basic properties
Nov 2 • The Cauchy and Cauchy-Goursat theorems
Nov 3 • Isolated singularities and winding numbers
• Cauchy integral formulas
• Holomorphic functions are infinitely differentiable
• Cauchy's inequality
• Liouville's theorem
• The Fundamental Theorem of Algebra
Nov 9 • The maximum modulus principle
Nov 10 NO CLASS: Midterm 2
Nov 16 • Morera's theorem
Chapter 5
• Sequences and series of complex numbers
• Taylor's theorem: statement, proof and examples
Nov 17 • Power series: convergence, continuity and holomorphicity
• Integrating and differentiating power series
• The principle of isolated zeroes
• The analytic continuation principle
• Towards Laurent series
Nov 23 • Laurent's theorem: statement, proof and examples
Chapter 6
• Isolated singularities
• Definition of residues
Nov 24 • The residue theorem
• Computing residues at poles
• Definite integrals involving trigonometric functions
Nov 30 • Improper integrals
Dec 1 • More improper integrals
Dec 7 • More improper integrals
Dec 8 Coming up…
• Inverse Laplace transforms