ℂ MATH 411: Honors Complex Variables

University of Alberta · Fall Term 2026

MATH 411: Honors Complex Variables

Complex numbers and analytic functions; Cauchy’s integral theorem and formula; the maximum modulus principle; Taylor and Laurent expansions; harmonic functions; residues and their applications; the argument principle; Rouché’s theorem; analytic continuation; and optional advanced topics such as normal families, the Riemann mapping theorem, and Picard’s theorem.

Instructor
Manish Patnaik
Email
patnaik@ualberta.ca
Office
UCOMM 5-254
Office hours
Fridays, 11:30 AM–12:30 PM in UCOMM 5-254 (or nearby)

Announcements

Class announcements

Office hours this Friday, October 23, will be from 2:00–3:00 PM.

Updated Lecture 18 notes are posted. A new remark explains that the proof of Theorem 3.1 only needs path connectedness; polygonal connectedness is not necessary. The standard proof for connected open sets actually gives the stronger conclusion of polygonal connectedness.

Earlier announcements

Homework 2 solutions are posted. Review materials for Midterm 1 (Friday, October 9) are posted under Midterm 1, including a coverage sheet and past exams with solutions.

Homework 2 is posted. It covers Lectures 6-8; the suggested completion date is September 28.

Lecture 6 posted. Office hours changed to 11:30-12:30pm on Fridays

Lecture 3 notes have been posted.

Lecture 2 notes have been posted.

Welcome to MATH 411! Lecture notes will be posted here after each class, and problem sets appear under Problem sets as the term goes on. Note that problem sets are not to be turned in.

I will use Canvas to send emails and to store your midterm and final exam grades. Other aspects of the course will be handled on this website.

Office hours will start on the second week of classes.

Lecture 1 notes have been posted. Homework 1 posted.

Syllabus

Course information

Meets
MWF, 3:00 – 3:50 PM
Location
CAB 369
Honors Complex Variables syllabus Official course syllabus View syllabus →

Required textbook: Serge Lang’s Complex Analysis, 4th edition. Available free for UAlberta students via the link here.

Grading

ComponentWeight
Midterm 125%
Midterm 225%
Final exam50%

Schedule & Notes

Lecture schedule

All section numbers refer to Serge Lang’s Complex Analysis, 4th edition.

LectureDateTopicNotes
1Wed, Sep 2Basics of complex numbers; polar form (§§1.1–1.2)PDF
2Fri, Sep 4Stereographic projection; plotting complex-valued functions (§§1.3–1.4)PDF
3Wed, Sep 9Limits, open sets, and the complex derivative (§§1.3–1.4)PDF
4Fri, Sep 11Total differentiability and the Cauchy–Riemann equations (§1.6)PDF
5Mon, Sep 14Cauchy–Riemann equations; conformal maps; Green’s theorem (§§1.6–1.7)PDF
6Wed, Sep 16Formal power series (§2.1)PDF
7Fri, Sep 18Radius of convergence; Hadamard’s criterion (§2.2)PDF
8Mon, Sep 21Hadamard’s criterion: proof and examples (§2.2)PDF
9Wed, Sep 23Formal versus convergent power series; analytic functions (§2.4)PDF
10Fri, Sep 25Analytic functions and their derivatives; isolated zeros and uniqueness of power series (§§2.3–2.5)PDF
11Mon, Sep 28Analytic extensions and the complex exponential; path integrals, parametrization, and the ML estimate (§§2.3, 3.2)PDF
12Fri, Oct 2Cauchy–Goursat theorem for rectangles and its subdivision proof (§3.3)PDF
13Mon, Oct 5Local primitives; Cauchy’s theorem on a disc (§3.3)PDF
14Wed, Oct 7The Gaussian integral and its Fourier transform; shifting contoursPDF
—Fri, Oct 9Midterm 1Details →
15Wed, Oct 14Indented contour; compactness and the chain-of-discs lemma (§3.4)PDF
16Fri, Oct 16Integration along continuous curves; local Cauchy integral formula preview; homotopy of curves (§§3.4–3.5)PDF
17Mon, Oct 19Homotopy form of Cauchy’s theorem: proof, null-homotopy, and contour deformation (§3.5)PDF
18Wed, Oct 21Homotopy Cauchy review; local Cauchy integral formula; comparison of circles; global primitives and complex logarithms (§§3.5–3.6)PDF
19Fri, Oct 23Logarithms of nonvanishing functions; logarithm laws and local series; holomorphic functions are analytic; Cauchy estimates and Liouville’s theorem (§§3.6–3.7)PDF
20Mon, Oct 26Morera’s theorem and applications; Laurent series (§§5.2–5.3)—
21Wed, Oct 28Laurent series, singularities, and residues (§§5.2–5.3)—
22Fri, Oct 30The residue formula (§6.1)—
23Mon, Nov 2Applications of the residue theorem—
24Wed, Nov 4The argument principle and applications—
25Fri, Nov 6Midterm 2Details →
26Mon, Nov 16Analytic isomorphisms—
27Wed, Nov 18Schwarz’s lemma and automorphisms of the unit disc (§7.2)—
28Fri, Nov 20Analytic isomorphisms; automorphisms of the upper half-plane—
29Mon, Nov 23No class—
30Wed, Nov 25Fourier transforms, inversion, and Poisson formulas I—
31Fri, Nov 27Fourier transforms, inversion, and Poisson formulas II—
32Mon, Nov 30Gamma and zeta functions; analytic continuation (Lang, Ch. 15)—
33Wed, Dec 2Analytic continuation; the zeta function and primes (Lang, Ch. 16)—
34Fri, Dec 4Prime number theorem I (Lang, Ch. 16)—
35Mon, Dec 7Prime number theorem II (Lang, Ch. 16)—

Homework

Problem sets

Practice problems to test your understanding. These are not to be turned in. Please attempt as many as you can!

SetCoversSuggested completion datePDFSolutions
HW 1Lectures 1–4, 1.1–1.6Sep 18PDFPDF
HW 2Lectures 6–8Sep 28PDFPDF
HW 3Lectures 13–20: contour integrals, homotopy, logarithms, Cauchy estimates, and Laurent seriesOct 26PDFNot yet posted

Assessment

Midterm 1

Held during the regular class period.

Date & time
Friday, October 9, 2026 · In class
Format
In class, closed book
Coverage
Coverage PDF
2026 exam and solutions
Exam statistics
Statistics and grading rubric PDF
Review materials

Assessment

Midterm 2

Held during the regular class period.

Date & time
Friday, November 6, 2026 · In class
Format
In class, closed book
Coverage
Lectures 1–24
Review materials
Posted closer to the exam

Assessment

Final exam

Details to be announced

Date & time
Set by the registrar — TBA
Format
TBA
Coverage
TBA
Review materials
Posted in the last week of term