18.04 | Spring 2018 | Undergraduate

Complex Variables with Applications


Course Meeting Times

Lectures: 3 sessions / week, 1 hour / session

Recitations: 1 session / week, 1 hour / session


18.02 Multivariable Calculus and 18.03 Differential Equations or 18.032 Differential Equations


Complex analysis is a beautiful, tightly integrated subject. It revolves around complex analytic functions. These are functions that have a complex derivative. Unlike calculus using real variables, the mere existence of a complex derivative has strong implications for the properties of the function.

Complex analysis is a basic tool in many mathematical theories. By itself and through some of these theories it also has a great many practical applications.

There are a small number of far-reaching theorems that we’ll explore in the first part of the class. Along the way, we’ll touch on some mathematical and engineering applications of these theorems. The last third of the class will be devoted to a deeper look at applications.

The main theorems are Cauchy’s Theorem, Cauchy’s integral formula, and the existence of Taylor and Laurent series. Among the applications will be harmonic functions, two-dimensional fluid flow, easy methods for computing (seemingly) hard integrals, Laplace transforms, and Fourier transforms with applications to engineering and physics.


There is no required textbook. We will use the following written resources:

Homework and Exams

There are nine problem sets, two one-hour in-class exams, and a three-hour final exam.


MIT has a culture of teamwork. We encourage you to work with study partners.

  • Collaboration on homework is encouraged.
  • You must write your solutions yourself, in your own words.
  • You must list all collaborators and outside sources of information.

Remember it it is a serious violation of academic integrity to copy an answer without attribution.


Problem sets 25%
Hour exams 40%
Final exam 35%

In addition, a passing grade on the final exam is required in order to pass the class.

Course Info

As Taught In
Spring 2018
Learning Resource Types
Exams with Solutions
Lecture Notes
Recitation Notes
Problem Sets with Solutions