---
title: Residue Theorem
description: The residue theorem is a powerful tool in complex analysis for evaluating line integrals of analytic functions. Learners will find residues at singularities and apply the theorem to calculate challenging real and complex integrals.
category: mathematics
subcategory: complex-analysis
difficulty: beginner, intermediate, advanced
url: /subject/residue-theorem
---

# Residue Theorem

The residue theorem is a powerful tool in complex analysis for evaluating line integrals of analytic functions. Learners will find residues at singularities and apply the theorem to calculate challenging real and complex integrals.

## Available Resources

1 Books • 3 Courses • 2 Websites

## Websites

### 1. Residue Theorem - Wikipedia

Encyclopedia article stating the residue theorem and its relation to Cauchy's integral theorem and formula, with worked examples evaluating real integrals and infinite series by contour integration. Readers see how residues at poles determine closed contour integrals.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://en.wikipedia.org/wiki/Residue_theorem

**Tags:** residue theorem, complex analysis, contour integration, cauchy integral formula

### 2. 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

## Courses

### 1. Complex Analysis

Learn the residue theorem and complex integration techniques with Dr. Bazett's Complex Analysis course!

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

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

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

### 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

## Books

### 1. Complex Variables and Applications

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

The standard applied text since 1948. Chapters four through seven build contour integrals from scratch, then spend a full chapter on residue applications: improper integrals, Jordan's lemma, indented paths, indentation around a branch point, and integration along a branch cut.

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

**Link:** https://www.amazon.com/dp/0073383171?tag=edmonddante07-20

**Tags:** complex-analysis, contour-integration, residue-theorem, branch-cuts, improper-integrals, textbook

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