---
title: Analytic Functions
description: Analytic functions are locally given by convergent power series and are infinitely differentiable. Learners will understand the properties of complex-differentiable functions, including conformal mapping, and how to apply them to physical systems.
category: mathematics
subcategory: complex-analysis
difficulty: beginner, intermediate, advanced
url: /subject/analytic-functions
---

# Analytic Functions

Analytic functions are locally given by convergent power series and are infinitely differentiable. Learners will understand the properties of complex-differentiable functions, including conformal mapping, and how to apply them to physical systems.

## Available Resources

2 Books • 1 Courses • 1 Websites

## Courses

### 1. Complex Analysis

Learn analytic functions and complex integration with Prof. Agarwal's Complex Analysis course. Explore key theorems and applications.

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

**Link:** https://nptel.ac.in/courses/111105115

**Tags:** complex-analysis, analytic-functions, complex-integration, cauchy-theorem

## Books

### 1. A First Course in Complex Analysis

**Author:** Matthias Beck, Gerald Marchesi, Dennis Pixton, Lucas Sabalka

Open textbook (CC BY 4.0, version 1.6) for a one-semester undergraduate course. It moves from complex differentiability and Cauchy's theorem to power series and the Residue Theorem, with over 250 exercises and SageMath prompts. Best free rigorous entry-level text; gives beginners a no-cost alternative to Conway (graduate level). A free PDF is available.

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

**Link:** https://matthbeck.github.io/complex.html

### 2. Functions of One Complex Variable

**Author:** John B. Conway

Conway's graduate text in the Springer GTM series, a standard first-year course reference. It proves the core theory rigorously, from analytic functions and Cauchy's theorem through the Riemann mapping theorem, Runge's theorem, analytic continuation, and harmonic and entire functions.

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

**Link:** https://www.amazon.com/dp/0387903283?tag=edmonddante07-20

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

## 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

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