---
title: Real & Complex Analysis
description: This subject studies the rigorous foundations of calculus, limits, and functions of real and complex variables. Learners will understand convergence, continuity, differentiability, and how to prove fundamental theorems of mathematical analysis.
category: mathematics
subcategory: real-analysis
difficulty: beginner, intermediate, advanced
url: /subject/real-and-complex-analysis
---

# Real & Complex Analysis

This subject studies the rigorous foundations of calculus, limits, and functions of real and complex variables. Learners will understand convergence, continuity, differentiability, and how to prove fundamental theorems of mathematical analysis.

## Available Resources

1 Books • 3 Courses • 3 Websites

## Websites

### 1. Analysis Wiki

LibreTexts shelf of free, openly licensed analysis textbooks, among them Trench's Introduction to Real Analysis, Zakon's Mathematical Analysis and Orloff's Complex Variables with Applications. Covers limits, continuity, differentiation, integration, infinite series and analytic functions, from introductory proof-based courses to graduate level.

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

**Link:** https://math.libretexts.org/Bookshelves/Analysis

**Tags:** real-analysis, complex-analysis, open-textbooks, proofs

### 2. Terry Tao's Blog: Harmonic Analysis Notes

**Author:** Terence Tao

Terence Tao's blog posts tagged harmonic analysis, including graduate lecture notes and research discussions on Fourier transforms, singular integrals, restriction theory, and related estimates. Readers with real analysis background gain the working tools and intuitions of modern harmonic analysis from a leading practitioner.

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

**Link:** https://terrytao.wordpress.com/tag/harmonic-analysis/

**Tags:** harmonic-analysis, fourier-analysis, singular-integrals, real-analysis, lecture-notes

### 3. complex-analysis.com

A focused online resource offering tutorials and notes on complex analysis. It covers holomorphic functions, Cauchy’s theorem, contour integration, residues, conformal mappings, and related topics, with worked examples and exercises.

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

**Link:** https://complex-analysis.com

**Tags:** websites, mathematics-statistics, mathematics

## Courses

### 1. 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

### 2. Real Analysis (MIT 18.100A)

**Author:** Casey Rodriguez

Rigorous foundations of calculus: real numbers, convergence of sequences and series, continuity, differentiability, the Riemann integral, sequences and series of functions, uniform convergence and interchange of limits. Includes 25 lecture videos, notes, problem sets and exams, and teaches how to read and construct proofs.

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

**Link:** https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/

**Tags:** real-analysis, proofs, sequences-and-series, uniform-convergence, riemann-integral

### 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 Analysis

**Author:** Lars V. Ahlfors

Ahlfors' classic graduate text on functions of one complex variable. Develops analytic functions, conformal mapping, Cauchy's integral theorem, series and product expansions, harmonic functions and the Riemann mapping theorem, giving readers the rigorous foundation expected in graduate analysis courses.

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

**Link:** https://www.amazon.com/dp/0070006571?tag=edmonddante07-20

**Tags:** complex-analysis, analytic-functions, conformal-mapping, cauchy-integral-theorem, riemann-mapping-theorem

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