---
title: Complex Numbers
description: Complex numbers extend the real number system by introducing the imaginary unit. Learners will understand algebraic operations, polar representation, Euler's formula, and how to solve polynomial equations that have no real roots.
category: mathematics
subcategory: complex-analysis
difficulty: beginner, intermediate, advanced
url: /subject/complex-numbers
---

# Complex Numbers

Complex numbers extend the real number system by introducing the imaginary unit. Learners will understand algebraic operations, polar representation, Euler's formula, and how to solve polynomial equations that have no real roots.

## Where to start

Start with Introduction to Complex Analysis, a free Coursera course from Wesleyan University taught by Petra Bonfert-Taylor, which covers complex numbers before analytic functions and contour integration. If you only use one resource, make it Visual Complex Analysis by Tristan Needham, a visual treatment of the subject. For the full theory, continue with Complex Variables with Applications (MIT 18.04) by Jeremy Orloff.

## Available Resources

2 Books • 3 Courses • 1 Websites

## Courses

### 1. Complex Analysis

Dive into complex numbers with Dr. Bazett's Complex Analysis course! Learn essential theory and problem-solving techniques.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PLBh2i93oe2qtIc75sLYaVEBt0QNqVbdmZ

**Tags:** complex-analysis, holomorphic-functions, complex-integration, cauchy-integral-formula

### 2. Complex Variables with Applications (MIT 18.04)

**Author:** Jeremy Orloff

Complex analytic functions and their applications: Cauchy's theorem and integral formula, Taylor and Laurent series, residues for evaluating hard integrals, harmonic functions, conformal maps, two-dimensional fluid flow, and Laplace and Fourier transforms. Provides 14 lecture note files plus problem sets and exams with solutions.

**Difficulty:** Intermediate | **Price:** Free

**Link:** https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/

**Tags:** complex-analysis, analytic-functions, residue-theorem, laurent-series, conformal-mapping

### 3. Introduction to Complex Analysis

**Author:** Petra Bonfert-Taylor

Wesleyan University course on complex analysis covering complex numbers, analytic functions, conformal mappings, Möbius transformations, contour integration, Cauchy's theorem, power and Laurent series, and residues. Learners finish able to evaluate contour integrals and analyze complex-differentiable functions.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://www.coursera.org/learn/complex-analysis

**Tags:** complex-analysis, analytic-functions, conformal-mapping, contour-integration, residue-theorem

## Websites

### 1. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Books

### 1. Complex Variables and Applications

**Author:** James Ward Brown, Ruel V. Churchill

Standard undergraduate textbook on complex variables for mathematics, physics and engineering students. Covers analytic functions, elementary functions, contour integrals, Cauchy's theorem, Taylor and Laurent series, residues and conformal mapping, teaching readers to evaluate real integrals and solve boundary-value problems.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0070108552?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, complex-analysis

### 2. Visual Complex Analysis

**Author:** Tristan Needham

Geometric introduction to complex analysis that explains results through pictures rather than heavy computation. Covers Möbius transformations, the complex derivative as an amplitwist, non-Euclidean geometry, winding numbers, Cauchy's theorem and vector fields, building intuition for why the standard theorems are true.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0198534469?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, complex-analysis

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