Math 185: Introduction to Complex Analysis Fall 2020

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

GSI: Aaron Brookner (find his e-mail on math.berkeley.edu)

Grader: Hugh Jiang

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

GSI Office hours: GSI Office hours are Tuesdays 9am-2pm, Wednesdays 9am-12pm, Thursdays 9am-12pm in 961 Evans

My Office hours: Will email them on Bcourses. Always feel free to send me questions or ask for alternative office hours.

Final exam: Friday, May 15, 2020. Location TBA (probably same as classroom but not guaranteed).

Prerequisites: Math 104 or equivalent. I will assume familarity with basic concepts like sup, inf, Cauchy sequences, etc. See this page for a useful summary of Math 104 material from another section of Math 185. In addition, basic knowledge of multivariable calculus is expected, including partial derivatives, line integrals, and Green's Theorem (see the bottom of Notes on mathematical proofs.

Text: The primary text for this course is Notes on Complex Analysis ([O]). Students should feel free to consult other books for additional exercises and/or alternative presentations of the material. Wikipedia also has lots of great articles on the topics at hand (e.g., see articles on Complex Analysis, Holomorphic Functions, Cauchy's Theorem, and many of the other concepts we will discuss in class). Students are expected to read the relevant sections of the notes, as the lectures are meant to complement the notes, not replace it, and we have a lot of material to cover.

Grading: Your homework grade (hw) will be the average of all homeworks, with the lowest dropped. Your exam grade (exams) 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. Finally, your total grade will be calculated as the maximum of: (0.2)hw + (0.8)exams, (0.3)hw + (0.7)exams.

Website: For now, the only website is this page, /~dcorwin/math185S2.html. I will use bcourses for solutions and other non-public information, such as my phone number.

Course policies:

Additional resources:

Course Overview: Outlined below is the rough course schedule. Depending on how the class progresses it may be subject to minor changes over the course of the semester.

Course calendar; "[O] refers to Orloff's notes, [G] refers to Gamelin's book, and [W] to Wikipedia.
Date Meeting Topics References Alternative References
1/22 1 Logistics/Syllabus, Overview of Course Overview of Complex Analysis, Topic 0 of [O] Complex Analysis on [W]
1/27 2 Complex Numbers Finish overview, Topic 1 of [O] Chapter 1 of [G], Complex Number, Complex Conjugate, Root of Unity, Euler's Formula on [W]
1/29 3 Finish Complex numbers (cont.) Finish Topic 1 of [O] Argument on [W], Chapter I of [G]
2/3 4 Begin analytic/holomorphic functions Begin Topic 2 of [O] Holomorphic Function on [W], Chapter II.1-4 of [G]
2/5 5 Analytic/holomorphic functions (cont.) Topic 2 of [O] Holomorphic Function on [W]
2/10 6 Line Integrals and Cauchy's Theorem Topic 3 of [O] Cauchy's Integral Theorem on [W], Chapter III.1-2 of [G]
2/12 7 Line Integrals and Cauchy's Theorem Topic 3 of [O] Cauchy's Integral Theorem on [W], Chapter III.1-2 of [G]
2/17 HOLIDAY
2/19 8 Cauchy's Theorem and Integral Formula Topic 4 of [O] Cauchy's Integral Formula on [W], Chapter IV.1-3 of [G]
2/24 9 Cauchy's Integral Formula (cont.) Topic 4 of [O] Chapter IV.4 of [G]
2/26 10 Harmonic Functions, Begin Fluid Flow Topic 5 of [O] Harmonic Function on [W], Chapter II.5 and III.3-5 of [G]
3/2 11 Fluid Flow and Complex Potentials (cont.), Begin Taylor Series Topic 6 of [O] Potential Flow on [W], Chapter III.6 of [G]
3/4 12 Taylor and Laurent Series Topic 7 of [O] Taylor Series, Zeroes and Poles on [W], Chapter V of [G]
3/9 13 Midterm 1 Review problems on Bcourses Test on [W]
3/11 14 Taylor and Laurent Series (cont.), Residue Theorem Topic 7 of [O], Topic 8 of [O] Laurent Series on [W], Chapter VI of [G]
3/16 15 Residue Theorem Topic 8 of [O] Residue Theorem on [W], Chapter VII.1 of [G]
3/18 16 Computing Definite Integrals Topic 9 of [O] Applications of the residue theorem to real integrals
3/23-3/27 Spring Break
3/30 Class cancelled Topic 9 of [O] Chapter VII.2-3 of [G]
4/1 17 Computing Definite Integrals (cont.) Topic 9 of [O] Chapter VII.2-3 of [G]
4/6 18 Computing Definite Integrals (cont.) Topic 9 of [O] Chapter VII.2-3 of [G]
4/8 19 Conformal Mappings Topic 10 of [O] Conformal Map on [W], Chapter II.6-7 of [G]
4/13 20 Conformal Mappings (cont.) Topic 10 of [O] Möbius Transformation on [W]
4/15 21 Argument Principle 11.1-11.2 of [O] Argument Principle on [W], Chapter VIII of [G]
4/20 22 Midterm 2 Review problems on Bcourses Test on [W]
4/22 23 Laplace Transform and Gamma Function 12.1, 12.3, 12.5, 13 of [O] Chapter XIV.1-2 of [G], Gamma Function on [W]
4/27 24 Analytic Continuation and the Gamma Function Topic 13.2 of [O], More on analytic continuation, Gamma sine formula Chapter V.8 of [G], Analytic Continuation on [W]
4/29 26 TBD (Elliptic functions? Riemann Zeta?) Riemann Zeta Function, Elliptic Functions Chapter XIV.3 of [G], Riemann Zeta Function on [W], Elliptic Functions on [W]
5/15 Final Exam

Homework and Exams: