MATH 106 – Functions of a Complex Variable | 2022-2023

This is the webpage for the course MATH 106 – Functions of a Complex Variable.

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Lecture times: Tuesday+Thursday 9:00 AM – 10:20 AM (Room: 380-380C).

Instructor: Jacopo Borga (jborga_at_stanford.edu) Office hours: Tuesday 11:00-12:00 AM (Room: 382-Q2).

Assistant: Judson Otto Kuhrman (kuhrman_at_stanford.edu) Office hours: Monday Wednesday (Room: ).

Description: Math 106 is an introductory course on complex analysis (focused on functions of a complex variable). We begin with complex numbers and basic topology of the complex plane. The course will cover analytic functions, Cauchy-Riemann equations, complex integration, Cauchy integral formula, residues, and elementary conformal mappings. By the end of the course you should be able to:

  • Illustrate the geometry of complex numbers and its behavior under transformations.
  • Demonstrate connections and distinctions between properties of real variable calculus and functions of a complex variable.
  • Characterize analytic functions of a complex variable and utilize their rigidity properties in multiple forms to study them.
  • Apply the theory of complex variable functions to other areas of mathematics and science. These applications may include real variable integration, differential equations, fluid flows, electromagnetic fields, and prime numbers.

Prerequisites: Math 52, and in particular a strong foundation in calculus, including: integration and partial derivatives; Taylor series and power series; parametrized paths and path integrals.

Program:  Here is a tentative program for the course (for each lecture I will provide my hand-written lecture notes):

Part A. Preliminaries: Complex numbers; Topology on C (convergent sequences; open, closed, compact and connected sets; continuous functions); More on convergence (convergent and absolutely convergent series; Cauchy product of absolutely convergent numerical series; sequences of functions; series of functions).

Part B. Power series and analytic functions: Power series (radius of convergence; sum and product of power series); Analytic functions (algebra of analytic functions in a domain; isolated zeros and analytic continuation; exponential and π).

Part C. Holomorphic Functions: Complex differentiability (definition and basic properties; Cauchy-Riemann equations); Holomorphy of analytic functions (power series; analytic functions;  Cauchy formula; analyticity of holomorphic functions); First main theorems on holomorphic functions (Cauchy’s inequalities; Liouville theorem; Open mapping theorem; Maximum modulus principle).

Part D. Path integrals: Basics (definition and examples; some computation rules; anti-derivative and integrals); Path integrals and holomorphic functions (holomorphy criteria via integrals; complex derivatives are automatically continuous; summary on holomorphic functions); Winding numbers (definition and properties; Cauchy formula with general paths in star-shaped domains; computation of winding number); General Cauchy formula (statement and proof; some direct corollaries); Homotopy and simply connected sets (homotopy and path integrals; simply connected sets; a connection to topology: the fundamental group); Complex logarithms and m-th roots  (logarithms; complex powers and m-th roots; an application: local normal form and biholomorphic function).

Part E. Isolated singularities and the residue theorem: Laurent’s expansions; Isolated singularities (removable singularity; poles; essential singularities; summary of isolated singularities; meromorphic functions); Residue theorem (statement; application to computation of real integrals; counting zeroes: the argument principle).

Part F. Applications: TBD

Week 1 (09/26): Part A

Week 2 (10/03): 

Week 3 (10/10): 

Week 4 (10/17)

Week 5 (10/24): 

Week 6 (10/31): *November 1 (Tue) –> Midterm exam at 9:00 AM -10:30 AM (on the material covered on Weeks 1-2-3-4)

Week 7 (11/07): *November 8 (Tue) –> Democracy Day: day of civic service – no class

Week 8 (11/14): 

*November 21-25 (Mon-Fri) Thanksgiving Recess (no classes)

Week 9 (11/28):

Week 10 (12/05):

Exercises: A new homework will be posted at the start of every week (or earlier), and should be submitted on Gradescope by 10PM the following Tuesday. I will set up Gradescope to actually accept submissions until a day later, as a grace period in case of last-minute technical difficulties. Any other delay will be not accepted (i.e., no grading for late submissions). Also, your lowest homework score will be dropped from consideration.

You are permitted (and encouraged!) to discuss the problems with other students, but you must write up the solutions yourself. Please work out problems neatly — do not hand in your scratch work.

Midterm exam: Tuesday, November 1 at 9:00 AM -10:30 AM (on the material covered on Weeks 1-2-3-4).

Final exam: Wednesday, December 14 at 8:30-11:30 AM (Room: ?).

Grading: Judgement based on Midterm exam mark (30%), Final exam mark (40%) and on consistent Homework (30%).

Textbook (NOT STRICTLY NEEDED): Stephen D. Fisher, Complex Variables, 2nd Edition. We will cover material from chapters 1, 2 and 3.

Additional info: Consistent with university policy, face coverings are required in classrooms until further notice. Face coverings may, however, be removed briefly while speaking. As Stanford strongly recommends masking indoors, face coverings should also be worn when attending office hours. Eating is not permitted in classrooms. Students should attend the in-person lecture and discussion section to which they are officially assigned. This ensures that classrooms do not exceed official room capacity, and supports prompt notification should a specific section need to move online at short notice. Course messages are sent out via Canvas, please ensure that your Canvas notifications are on so that you receive any announcements promptly. As standard practice, lectures and discussion sections in Mathematics courses are taught in-person. As such, Zoom links will not be provided. Additionally, in-person lectures and discussion sections will not be recorded. Students who miss class due to illness (including COVID-19) should make arrangements to obtain lecture notes from other students in the class. As standard practice, there are no make-up exams or remote exams. If you will miss an exam due to illness, please reach out to your instructor for more information.