Syllabus

In this course, we will learn about one-variable complex analysis. This is the study of complex functions of one complex variable. It is simpler and more elegant than the study of functions of two real variables, and in some ways even simpler than the study of functions of one real variable. For instance, a complex function with one continuous first derivative automatically has an infinite number of derivatives, so a whole class of pathological examples in real analysis have no analogues in the complex case.

There is a final paper for the course. For details on logistics ad suggestions on topics, please see this handout. A problem statement is due on November 11, a first version is due on November 23, and the final paper is due on December 3.

For more, see the complete syllabus.

Course schedule

Date Material References
Sep. 28
  • Definition of line integrals
  • Elementary properties
  • First connection with Cauchy-Riemann
Bak–Newman, 4.1
Notes, 4.1–4.3
Sep. 30
  • Integrals and anti-derivatives
  • Rectangle theorem
Bak–Newman, end of 4.1 and 4.2
Notes, 5.1–5.3
Oct. 5
  • Green's theorem and integration of holomorphic functions
  • Determining functions by boundary values: Intro to Fourier analysis
Notes, 5.4–5.5
For an intro to Fourier analysis, see Chapters 1 and 2 of Fourier Analysis: An Introduction by Stein and Shakarchi.
Oct. 7
  • Cauchy's integral formula
  • Power series representation for holomorphic functions
Bak–Newman, 5.1
Also Stein–Shakarchi, 2.4, pp. 45–48
Oct. 12
  • Schwarz's Lemma and Cauchy's Inequalities
  • Liouville's Theorem
  • Fundamental Theorem of Algebra
Stein–Shakarchi, 2.4, pp. 48–53
Bak–Newman, 5.2
Oct. 14
  • Power series for non-entire functions
  • Analytic continuation
  • Uniqueness Theorem
  • Maximum modulus theorem and relatives
Bak–Newman, 6
Oct. 19 Local behaviour of holomorphic functions
  • Order of zeroes
  • Functions growing at infinity
  • Singularities: Removable, poles, and essential
  • Criteria for singularities
Bak–Newman, 9.1
Stein–Shakarchi, 3.1
Notes, Section 6
Oct. 21
  • Casorati-Weierstrass Theorem
  • Meromorphic functions
  • Riemann sphere
Bak–Newman, 9.1, 1.4
Needham, 3.IV (see also 3.II)
Stein–Shakarchi, 3.3, 1.4
Oct. 26
  • Möbius transformations
Möbius Transformations Revealed, movie and article
Needham, 3.I, 3.V
Oct. 28
  • Paper topics
Paper description handout
Nov. 4
  • Homotopy of curves
  • Simply-connected regions
  • Winding number
Notes, Section 7
Stein–Shakarchi, 3.5
Nov. 9
  • Laurent expansions
  • Evaluation of integrals
Bak–Newman, 9.2, 10.1 Stein–Shakarchi, 2.3
Nov. 11
  • More integrals
  • Counting zeroes and poles
Bak–Newman, 10.2, 11.1
Nov. 16
  • Evaluating sums
  • Zeroes of entire functions
Bak–Newman, 11.2 Stein–Shakarchi, 5.1
Nov. 18
  • Functions of finite order
  • Function engineering
Stein–Shakarchi, 5.2, 5.3
Nov. 23
  • Conformal mapping examples
Stein–Shakarchi, 5.4
Stein–Shakarchi, 8.1, 8.2
Nov. 23
  • Riemann mapping theorem
  • Hyperbolic geometry
Stein–Shakarchi, 8.3

Homeworks

Lecture notes

Lecture notes (Updated: Nov. 4)

Please note that these are fragmentary in places, often just enough to remind me of my thinking, but they may still be helpful for you. For material that is not covered in the textbook(s), I will make an effort to make the notes more complete.

Midterm

Here is the midterm, which was due Oct. 26.

See also the CourseWorks page.