This is the non-bcourses version of the webpage for Math 185, lecture 4, as offered in Fall 2025. Note that although some materials may be common to other versions of the class, e.g. my Spring 2026 section, others will not be, and policies may differ.
The syllabus can be found here. Lecture notes, homework, and other materials can be found below.
Instructor: Avi Zeff
Time/place: Monday, Wednesday, and Friday 9-10 AM, in Etcheverry 3111
Office hours: Monday, Wednesday, and Friday 10-11 AM, in 860 Evans Hall
Lecture 1: introduction, complex numbers and representations
Lecture 2: stereographic projection and first branch cuts
Lecture 3: the logarithm and power functions
Lecture 4: trigonometric functions and notions from analysis
Lecture 5: the Cauchy-Riemann equations
Lecture 6: harmonic functions and conformal mappings
Lecture 7: fractional linear transformations
Lecture 9: harmonic functions, reprise
Lecture 10: complex line integrals
Lecture 11: using Cauchy's formula
Lecture 12: applications of Cauchy's formula to analyticity
Lecture 14: series and convergence
Lecture 17: series expansions at infinity and other properties
Lecture 18: analytic continuation
Lecture 19: Laurent expansions
Lecture 20: isolated singularities
Lecture 21: the residue theorem
Lecture 22: applications of the residue theorem
Lecture 23: the argument principle
Lecture 25: conformal maps to the disk
Lecture 26: the Riemann mapping theorem
Lecture 27: harmonic functions and the Poisson integral formula
Practice problems for midterm 1
Practice problems for midterm 2
Practice problems for midterm 3