Math 185: Introduction to Complex Analysis

Instructor: David Corwin (dcorwin at berkeley dot edu). In all e-mail correspondence, please include "[Math185]" in the subject line.

GSI: Benjamin Filippenko (river at berkeley dot edu)

Lecture: TTh 5pm-6:30pm in Hearst Mining 310

GSI Office hours: GSI Office hours are in Evans 961. They are 3-5pm every day of the week.

My Office hours: Location: Evans 749 RRR Office hours: 3:30-5 on Tuesday, Dec 4. 2-3:30 and 5-6 on Wednesday, Dec 5. Please e-mail me if you'd like other office hours.

Final exam: Thursday, 12/13/18 11:30am-2:30pm. Location: classroom.

Prerequisites: Math 104 or equivalent. I will assume familarity with basic concepts like sup, inf, Cauchy sequences, etc. In addition, basic knowledge of multivariable calculus is expected, including partial derivatives and line integrals.

Text: The primary text for this course is Complex Analysis by Stein and Shakarchi [S-S]. Students should feel free to consult other books for additional exercises and/or alternative presentations of the material (see in particular the book by Gamelin [G] linked below, which is available electronically to all UCB students). Students are expected to read the relevant sections of the textbook, as the lectures are meant to complement the textbook, not replace it, and we have a lot of material to cover.

Grading: 20% homework, 2 x 20% in-class midterms (10/2 and 11/8), 40% final exam. The lowest two homework scores will be dropped. No makeups for the midterms will be given except in cases requiring special accommodation. Your exam grade will be computed based on the maximum of the following three schemes: (0.2)MT1 + (0.2)MT2 + (0.4)F; (0.2)MT1 + (0.6)F; (0.2)MT2 + (0.6)F

Website: For now, the only website is this page, /~dcorwin/math185F18.html. I will use bcourses for solutions and other non-public information, such as my phone number.

Course policies:

Homework will be assigned regularly (see the syllabus) and due at the beginning of class. I grant extensions in reasonable circumstances, but you must talk to me as early as possible. The longer you wait, the less flexible I will be.
You may work together to figure out homework problems, but you must write up your solutions in your own words in order to receive credit. In particular, please do not copy answers from the internet or solution manuals. Since a major purpose of the homework is to prepare you for the exams, I encourage you to give each problem an honest shot by yourself (say, at least thirty minutes) before discussing it with others. Another useful practice is if you're stuck on a problem, come to office hours and ask for a hint. The more you figure out on your own, the better your understanding of the material, and the better you'll do both on the exams and in your future endeavours that might require complex analysis.
The usual expectations and procedures for academic integrity at UC Berkeley apply. Cheating on an exam will result in a failing grade and will be reported to the University Office of Student Conduction. Please don't put me through this.
Please let me know sooner rather than later if you need any accommodations related to the Disabled Student Program (DSP). I am more than happy to make arrangements, but it really helps if you tell me earlier rather than later.
Per university guidelines, it is your responsibility to notify the instructor in writing by the end of the second week of classes (September 7) of any scheduling conflicts due to religious observance or extracurricular activities, and to propose a resolution for those conflicts.

Additional resources:

Complex Analysis by Theodore W. Gamelin (accessible from UCB IP addresses)
Complex Analysis by George Cain (a personal favorite, gives a nice overview of the basic results of complex analysis)
Class Notes from 18.04 by Jeremy Orloff
For multivariable calculus, I like: Multivariable Calculus by George Cain and James Herod

Course Overview: The goal of this course is to introduce students to the world of complex analysis. On the face of it, complex analysis is just differentiating and integrating with respect to a complex variable rather than a real variable. However, the two-dimensional nature of the complex numbers gives complex analysis many interesting features unknown to students of real analysis. The bulk of the course will consist in developing the basics of complex analysis, roughly Chapters 1-3 of [S-S]. After that, I hope to cover a few additional topics, depending on time and the interests of the students. Possiblities include the gamma function, the Riemann zeta function, conformal mappings, the Hadamard product formula, elliptic functions, and modular forms.

Course calendar; "[S-S]" and "[G]" refer to Stein-Shakarchi and Gamelin, respectively.
Date	Meeting	Topics	References	Comments
8/23	1	Logistics/Syllabus, Overview of Complex Analysis	[S-S] Introduction, bits of Chp 1 Section 1.1, 2.2, Overview of Complex Analysis
8/28	2	Overview (cont.), Complex Numbers, Limits, and Topology	[S-S] Ch 1 Sections 1.1-1.3, 2.1	Homework 1 put online, due 9/6
8/30	3	Holomorphic functions, Cauchy-Riemann eqns, examples	[S-S] Ch 1 Sections 2.1-2.2
9/4	4	Finish C-R Equations, some elementary functions, power series	[S-S] Chp 1 Section 2.2-2.3
9/6	5	Curves in the plane, Contour integration	[S-S] Ch 1 Section 3	HW 1 Due
9/11	6	Contour integration (cont.), Primitives and Path Independence, Begin Goursat's Theorem if time	[S-S] Ch 1 Section 3
9/13	7	Goursat's Theorem	[S-S] Ch 2 Section 1	HW 2 Due
9/18	8	Consequences of Goursat's Theorem	[S-S] Ch 2 Sections 1 and 2, Interiors and Simple-Connectivity
9/20	9	Evaluation of Some Integrals	[S-S] Ch 2 Section 3
9/25	10	Cauchy's integral formula and consequences	[S-S] Ch 2 Section 4	HW 3 Due
9/27	11	Cauchy's integral formula and consequences, (cont.)	[S-S] Ch 2 Section 4
10/2	12	Midterm 1
10/4	13	Analytic Continuation, Morera's Theorem and consequences	[S-S] Ch 2 Sections 4 and 5,	HW 4 Due
10/9	14	Morera's Theorem and consequences	[S-S] Ch 2 Sections 5.1-5.2
10/11	15	More consequences of Morera's Theorem	[S-S] Ch 2 Section 5.3
10/16	16	More consequences of Morera's Theorem	[S-S] Ch 2 Section 5.3-5.4	HW 5 Due
10/18	17	Poles and Zeroes	[S-S] Ch 3 Section 1
10/23	18	Poles and Residues	[S-S] Ch 3 Section 1
10/25	19	The Residue Theorem	[S-S] Ch 3 Section 2
10/30	20	The Residue Theorem	Proof of the Residue Theorem	HW 6 Due
11/1	21	Isolated Singularities and Meromorphic Functions	[S-S] Ch 3 Section 3
11/6	22	Meromorphic Functions, Casorati-Weierstrass Theorem	[S-S] Ch 3 Section 3	HW 7 Due
11/8	23	Midterm 2	Study!
11/13	24	Meromorphic Functions at infinity, Riemann Sphere, Begin Argument Principle if time	[S-S] Ch 3 Section 3-4
11/15	25	Argument Principle, Rouche's Theorem, Open Mapping Theorem, Maximum Principle	[S-S] Ch 3 Section 4
11/20	n/a	Class was cancelled due to smoke	[S-S] Ch 3 Sections 5-6
11/27	26	Simply Connected Regions	[S-S] Ch 3 Section 5-6	HW 8 Due
11/29	27	The Complex Logarithm and Multi-Valued Functions	[S-S] Ch 3 Section 6	Wikipedia on the Riemann surface of log (with picture) MIT OCW on Multi-valued Functions and Branches A video showing what happens for the square root
12/4	28	The Gamma and Zeta Functions	[S-S] Ch 6 Section 1 up to and including Theorem 1.4 (no Theorem 1.6 or 1.7), Section 2 Propositions 2.1 and 2.5 (ignore everything about the theta function)	HW 9 Due
12/13/18		Final Exam		.

Homework and Exams:

Homework 1, due Thursday 9/6. Solutions.
Homework 2, due Thursday 9/13. Solutions.
Homework 3, due Tusday 9/25. Solutions.
Homework 4, due Thursday 10/4. Solutions.
Midterm 1 Study Guide and Links to Practice Problems. Also see Midterm 1 Note Sheet.
Midterm 1 Solutions
Homework 5, due Tuesday 10/16. Solutions.
Homework 6, due Tuesday 10/30. Solutions.
Homework 7, due Tuesday 11/6. Solutions.
Midterm 2 Study Guide and Links to Practice Problems. Also see Midterm 2 Note Sheet.
Homework 8, due Tuesday 11/20. Solutions.
Homework 9, due Tuesday 12/4. Solutions.
Final Draft Notesheet. Also more practice problems from material after midterm 2 and (partial) solutions to these practice problems.