## Course Summary

Most culture's mathematics begins with the introduction of the counting numbers and their structure under addition, with multiplication following soon thereafter. In western mathematics these two operations (together with the usual comparisons) were the language in which mathematics was written for thousands of years. The inverses of these operations were also considered, though ``improper fractions" were socially accepted long before negative numbers were thought to have real meaning. Though one might expect the acceptance of negative numbers to far predate that for complex numbers, the two were finally considered ``natural'' around the same time when it was discovered that there are integer-coefficient cubic equations with real (positive) roots that nonetheless require square roots of negative numbers to be formulated (you might look up casus irreducibilis if you want to find out more about this). This was the first in a long line of discoveries that pointed to the fact that the nature of the complex numbers is important to understand, even if you think you only care about real phenomena.

The goal of this class is to learn what we can about functions of a single complex variable. In many ways this will mean attempting to translate familiar notions from calculus into the complex setting, and we'll see that there are certain surprises in the study of complex functions which one might not have predicted from studying real calculus. A constant theme in the course will be connecting the geometric understanding of the complex numbers with the algebraic operations we perform on them. By the end of the course we will understand the classical results on derivatives and path integrals for complex functions, and we will (hopefully) discuss one of the most important complex functions in all of mathematics: the Riemann zeta function.

## Course Instructor

The professor for this class is Andy Schultz. His office is on the third floor of the Science Center, room S352. His office hours will be Monday from 3-4, Tuesday from 2:45-3:45, Wednesday from 9-10, Thursday from 10-11 and Friday from 9:30 to 10:30. You are highly encouraged to attend office hours, and you never need an appointment to do so. If these office hours don't fit with your schedule, contact the instructor so that he can either adjust when ``official" office hours are held or set up an appointment to help you outside of office hours.