Functions of a Complex Variable (SP24)

Course Info:

  • Lectures: Monday, Wednesday, Friday, block 9L (8:50 - 9:55 am ET)
  • x-period: Thursday, block 9LX (9:05 - 9:55 am EDT); used for biweekly quizzes
  • Room: Kemeny 006
  • Instructor: Andrew Hanlon
  • Instructor office: Kemeny 320
  • E-mail: andrew.hanlon@dartmouth.edu
  • Office hours: Monday 10 - 11 am, Wednesday 11am - 12pm, Thursday 10 - 11 am, and by appointment
  • Prerequisite: Math 13.  If you are unsure about your preparation, please talk to the instructor!
  • Course textbook: E.B. Saff and A.D. Snider, Fundamentals of Complex Analysis, 3rd edition. References in the schedule below are to this book. See the bottom of this page for additional resources.
  • Grading: Grades will be based on weekly homework (30%), biweekly quizzes (40%), and a final exam (30%). Your lowest homework score will be dropped.

Final exam:

Our final exam is on Saturday, June 1, 2023 at 3 pm ET in Kemeny 006. You can use one 8.5 x 11" double-sided page of notes on the exam. Here is a practice exam and solutions. Here is the final exam and solutions.

Quizzes:

We will have quizzes every other week during the x-period according to the schedule below. Each quiz concerns material covered in class up to and including the Monday before the quiz.

Date Practice Quiz Quiz
4/4 Practice Quiz 1, Solutions Quiz 1, Solutions
4/18 Practice Quiz 2, Solutions Quiz 2, Solutions
5/2 Practice Quiz 3, Solutions Quiz 3, Solutions
5/16 Practice Quiz 4, Solutions Quiz 4, Solutions

Homework:

Homework is due on Gradescope on the day indicated below before 11:59 pm ET. Please upload your solutions by following the instructions in this video.  Feel free to submit handwritten solutions or work typeset in LaTeX (Overleaf is a great resource and tool for the latter). If you are skipping a problem, it is easier for the grader if you write "Skip" and upload that as your solution to the problem. You are more than welcome to work with others on the homework problems. However, the solutions should be written up on your own and reflect your understanding.

Due Date Homework Solutions
3/26 Homework 0
4/2 Homework 1, tex HW1 Solutions
4/9 Homework 2, tex HW2 Solutions
4/16 Homework 3, tex HW3 Solutions
4/23 Homework 4, tex HW4 Solutions
4/30 Homework 5, tex HW5 Solutions
5/7 Homework 6, tex HW6 Solutions
5/14 Homework 7, tex HW7 Solutions
5/21 Homework 8, tex HW8 Solutions
5/28 Homework 9, tex HW9 Solutions

Schedule:

The specific topics covered in each class are tentative and will be updated as the semester progresses.  Let me know if you find any errors in the posted notes.

Date Topic References
M 3/25 Why complex analysis? Complex algebra §1.1, Notes
W 3/27 Visualizing the complex plane §1.2, 1.3, Notes
F 3/29 Complex exponential and powers §1.4, 1.5, Notes, a fun article
M 4/1 Domains §1.6, Notes
W 4/3 Functions and continuity §2.1, 2.2, Notes
F 4/5 Complex derivative, CR equations §2.3, 2.4, Notes
M 4/8 Harmonic functions, Julia and Mandelbrot sets §2.5, 2.7, Notes, a fun article and pictures
W 4/10 Complex polynomials §3.1, Notes
F 4/12 Rational functions §3.1, 1.7, Notes
M 4/15 Trig functions and complex logarithm §3.2-3.5, Notes
W 4/17 More on log §3.3-3.5, Notes
F 4/19 Contour integrals and path-independence §4.1-4.3, Notes
M 4/22 Cauchy's integral formula §4.4b, 4.5, Notes
W 4/24 Continuity of f'(z) and infinite differentiability §4.5, Notes
F 4/26 Liouville's theorem and maximum principle §4.6, 4.7, Notes
M 4/29 Power series §5.1-5.4, Notes
W 5/1 More on power series §5.1-5.4, Notes
F 5/3 Zeroes and Laurent series §5.5, 5.6, Notes
M 5/6 Laurent series expansion on annulus §5.5, Notes
W 5/8 Singularity types, meromorphic functions §5.6, Notes
F 5/10 Infinity, analytic continuation §5.7, 5.8, Notes
M 5/13 Fourier series §8.1, Notes and a fun article
W 5/15 Residue theorem and applications §6.1-6.4, Notes
F 5/17 Keyhole integration §6.5, 6.6, Notes
M 5/20 Rouche's theorem and open mapping principle §6.7, Notes
W 5/22 Conformal maps and biholomorphisms §7.2, Notes
F 5/24 Möbius transformations §7.3, 7.4, Notes
W 5/29 Review Notes

Course Catalogue Description:

This course covers the differential and integral calculus of complex variables including such topics as Cauchy's theorem, Cauchy's integral formula and their consequences; singularities, Laurent's theorem, and the residue calculus; harmonic functions and conformal mapping. Applications will include two-dimensional potential theory, fluid flow, and aspects of Fourier analysis.

Learning Outcomes:

By the end of this course, you should be able to:

  1. Understand of the basic structures of the complex numbers and holomorphic functions: define terms, explain their significance, and apply them in context;
  2. Solve mathematical problems: utilize abstraction and think creatively; 
  3. Write clear mathematical proofs: recognize and construct mathematically rigorous arguments.

Additional Pages:

  1. Academic Honor Principle
  2. Expectations
  3. Student Accessibility Needs
  4. Mental Health
  5. Title IX
  6. Religious Observances
  7. Additional Support for your Learning

Additional Resources: 

  • https://complex-analysis.com/ has very cool visualizations of what we will be learning. There are also a few 3Blue1Brown videos on complex analysis topics.

  • Previous Math 43 webpages are available on the math department webpage. These may be another good source of exercises and practice exams.

  • It is highly recommended to solve as many exercises from the textbook or other resources as you have time for. This will be more useful than rereading your class notes or textbook as it will often force you to go back to look at the material anyway. The only way to truly learn mathematics is through practice. Feel free to ask about your solutions.

  • R. Hammack, Book of Proof is a free online textbook that you may find useful to sharpen your proof-writing skills.

  • There are tons of other great textbooks and online materials on complex analysis that you can consult. For instance, you can see the wide range of opinions on this StackExchange question.