# 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: 380-380X).

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:  (References to the textbook “Stephen D. Fisher, Complex Variables, 2nd Edition”)

Week 1 (09/26):

• Introduction and Complex Numbers; Geometry of the Complex Plane (§1.1-1.2)
• Functions of a Complex Variable, Roots (§1.4)

Week 2 (10/03):

• Open and Closed Sets; Limits and Continuity (§1.3-1.4)
• Analytic Functions and the Cauchy-Riemann Equations (§2.1)

Week 3 (10/10):

• Exponential and Trigonometric Functions, Logarithms and Branches (§1.5)
• Trigonometric Functions properties and Examples (§1.5)

Week 4 (10/17)

• Line Integrals and the Fundamental Theorem of Calculus (§1.6)
• Cauchy-Riemann Equations and Harmonic Functions; Harmonic Conjugates (§2.1)

Week 5 (10/24):

• Path Independence and Cauchy’s Formula (§2.3)
• Cauchy’s Formula and its applications (§2.3)

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)

• Power Series and Radius of Convergence (§2.2) Properties of Holomorphic Functions (§2.4)

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

• Liouville’s Theorem (§2.4) Singularities and Poles (§2.5)
• Residues and Laurent Series (§2.5)

Week 8 (11/14):

• The Residue Theorem and Applications (§2.6)
• Zeros of Analytic Functions (§3.1)

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

Week 9 (11/28):

• The Fundamental Theorem of Algebra and Rouche’s Theorem (§3.1)
• Maximum Modulus and Mean Value (§3.2)

Week 10 (12/05):

• Moebius Transformations (§3.3)
• Conformal Mappings (§ 3.4)

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.