Math 106: Functions of a Complex Variable
Spring 2026
Lectures: Tue, Th 9am-10:20am, room 380C.
Instructor: Eleny Ionel, office 383L, ionel "at" math "dot" stanford "dot" edu
Office Hours: Monday, 1-2pm, Tuesday 2-3pm and by appointment.
Course Assistant: Zhuo Zhang, office 381K, zhuozh "at" stanford "dot" edu
Office Hours: Monday 10:30 - 11:30 and Wed 10:30 - 12:30.
Course Description and Learning Goals: 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; Green theorem; parametrized curves and line integrals. Some basic familiarity with complex numbers (at the level of §1.1 in Fisher) would be helpful.
Textbook: Stephen D. Fisher, Complex Variables, 2nd Edition. We will cover material from chapters 1-3.
Students are expected to read the relevant sections of the textbook(s) (listed below) and any handouts (posted on Canvas) before coming to class each day.
Course Policies: Please see https://goto.stanford.edu/mathcoursepolicies for important course policies on exam conflicts, academic accommodations, AI guidance, and taking exams.It is your responsibility to thoroughly read this information.
Accommodations & Flexibility Form: http://goto.stanford.edu/math106oae
Course Logistics: Course announcements, homework and other course materials will be posted on Canvas. Weekly homework will be due and graded on Gradescope.
Homework: Weekly homework assignments given out on Wednesday, and due the following Wednesday at 11:59PM on Gradescope (unless otherwise noted). No late submissions will be accepted under any circumstances. (This is as much a courtesy to the grader as an incentive to stay current with the course and not fall behind.) However, to accommodate such as a serious illness or anything else that may arise, your homework score will be multiplied by 10/9 (not to exceed 100%) at the end of the quarter.
We encourage you to form study groups to discuss and work together on the homework (ideally after thinking about it by yourself). However, you must write up your own solutions individually and in your own words, and indicate the names of any collaborators or sources of outside help that you received. Keep in mind that simply copying solutions from another student or from other sources (such as internet or AI generated) and then submitting it for credit will be considered a violation of the Stanford Honor Code.
Exams: There will be two in-person, proctored exams: a midterm and a final exam.
Grading: The course grade is based on the following components:
Homework: 15%
Midterm Exam: 35%
Final Exam: 50%
Important dates:
- Final Study List deadline: Friday, April 17, 5pm.
- Midterm Exam: Thursday, May 7, in class.
- Final Exam: Monday June 8, 2026, 8:30am - 11:30am.
Tentative Schedule: (which may be adjusted as the quarter goes on)
- Week 1: Brief intro: Geometry of the Complex Plane, Functions of a Complex Variable (§1.1-§1.4)
- Week 2: Holomorphic functions and Cauchy-Riemann Equations; Harmonic functions and Harmonic Conjugates (§2.1);
Power Series and Radius of Convergence (§2.2);
- Week 3:
Exponential and Trigonometric Functions; Logarithms and Branches (§1.5);
Line Integrals and the Fundamental Theorem of Calculus (§1.6)
- Week 4:
Path Independence and Cauchy's Formula (§2.3)
Cauchy's Formula and Applications (§2.3);
- Week 5:
Properties of Holomorphic Functions (§2.4); Liouville's Theorem (§2.4)
- Week 6:
Singularities and Poles (§2.5);
midterm on Thursday
- Week 7:
Residues, Laurent Series and Applications (§2.5-§2.6)
- Week 8:
Zeros of Analytic Functions and Rouche's Theorem (§3.1);
Maximum Modulus and Mean Value (§3.2)
- Week 9:
Moebius Transformations (§3.3); Conformal Mappings (§ 3.4)
- Week 10: Applications
Academic Integrity:
The Honor Code articulates Stanford University's expectations of students and faculty in establishing and maintaining the highest standards in academic work. Its purpose is to uphold a culture of academic honesty. Students will support this culture of academic honesty by neither giving nor accepting unpermitted academic aid in any work that serves as a component of grading or evaluation, including assignments, examinations, and research. Examples of conduct that have been regarded as being in violation of the Honor Code (and are most relevant for this course) include copying from another student's work (or other sources such as the internet, AI generated etc) or allowing another student to copy from your own work; plagiarism; representing as one's own work the work of another. Please visit the OCS website for more information on the Honor Code, and please see link above for the Math Department AI Policy.