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 Download practice exam and solutions Download solutions. Here is the final exam Download final exam and solutions Download 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.
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 Links to an external site.. Feel free to submit handwritten solutions or work typeset in LaTeX (Overleaf Links to an external site. 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.
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 Download Notes |
| W 3/27 | Visualizing the complex plane | §1.2, 1.3, Notes Download Notes |
| F 3/29 | Complex exponential and powers | §1.4, 1.5, Notes Download Notes, a fun article Links to an external site. |
| M 4/1 | Domains | §1.6, Notes Download Notes |
| W 4/3 | Functions and continuity | §2.1, 2.2, Notes Download Notes |
| F 4/5 | Complex derivative, CR equations | §2.3, 2.4, Notes Download Notes |
| M 4/8 | Harmonic functions, Julia and Mandelbrot sets | §2.5, 2.7, Notes Download Notes, a fun article Links to an external site. and pictures Links to an external site. |
| W 4/10 | Complex polynomials | §3.1, Notes Download Notes |
| F 4/12 | Rational functions | §3.1, 1.7, Notes Download Notes |
| M 4/15 | Trig functions and complex logarithm | §3.2-3.5, Notes Download Notes |
| W 4/17 | More on log | §3.3-3.5, Notes Download Notes |
| F 4/19 | Contour integrals and path-independence | §4.1-4.3, Notes Download Notes |
| M 4/22 | Cauchy's integral formula | §4.4b, 4.5, Notes Download Notes |
| W 4/24 | Continuity of f'(z) and infinite differentiability | §4.5, Notes Download Notes |
| F 4/26 | Liouville's theorem and maximum principle | §4.6, 4.7, Notes Download Notes |
| M 4/29 | Power series | §5.1-5.4, Notes Download Notes |
| W 5/1 | More on power series | §5.1-5.4, Notes Download Notes |
| F 5/3 | Zeroes and Laurent series | §5.5, 5.6, Notes Download Notes |
| M 5/6 | Laurent series expansion on annulus | §5.5, Notes Download Notes |
| W 5/8 | Singularity types, meromorphic functions | §5.6, Notes Download Notes |
| F 5/10 | Infinity, analytic continuation | §5.7, 5.8, Notes Download Notes |
| M 5/13 | Fourier series | §8.1, Notes Download Notes and a fun article Links to an external site. |
| W 5/15 | Residue theorem and applications | §6.1-6.4, Notes Download Notes |
| F 5/17 | Keyhole integration | §6.5, 6.6, Notes Download Notes |
| M 5/20 | Rouche's theorem and open mapping principle | §6.7, Notes Download Notes |
| W 5/22 | Conformal maps and biholomorphisms | §7.2, Notes Download Notes |
| F 5/24 | Möbius transformations | §7.3, 7.4, Notes Download Notes |
| W 5/29 | Review | Notes Download 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:
- Understand of the basic structures of the complex numbers and holomorphic functions: define terms, explain their significance, and apply them in context;
- Solve mathematical problems: utilize abstraction and think creatively;
- Write clear mathematical proofs: recognize and construct mathematically rigorous arguments.
Additional Pages:
- Academic Honor Principle
- Expectations
- Student Accessibility Needs
- Mental Health
- Title IX
- Religious Observances
- Additional Support for your Learning
Additional Resources:
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https://complex-analysis.com/ Links to an external site. has very cool visualizations of what we will be learning. There are also a few 3Blue1Brown videos Links to an external site. on complex analysis topics.
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Previous Math 43 webpages are available on the math department webpage. These may be another good source of exercises and practice exams.
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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 Links to an external site. is a free online textbook that you may find useful to sharpen your proof-writing skills.
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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 Links to an external site..