Essentials of Complex Analysis: A Computational Approach
Marco Saragnese
Essentials of Complex Analysis is an introduction to the fundamental concepts in complex analysis, suitable for a first course at the undergraduate level. The book covers the elementary functions, the Cauchy–Riemann equations, complex integration, Cauchy's theorem and the residue theorem. It ends with a series of applications of complex analysis to hydrodynamics, transcendental equations, elliptic functions and more.
Each chapter is accompanied by exercises and solutions, and the text concludes with a review of the material and a sample final exam. Students can download a free, interactive ebook version to engage with live Wolfram Language code.
Additional materials, including videos, interactive demonstrations, quizzes and exams can be found online in the companion Wolfram U course.
Notebooks can be opened in Mathematica or the free Wolfram Player.
Information & Media Inquiries
- Title: Essentials of Complex Analysis: A Computational Approach
- Author: Marco Saragnese
- Paperback: forthcoming
- Kindle: forthcoming
- Wolfram Notebooks: download free
- Publisher: Wolfram Media, Inc.
- Publication Date: December 16, 2025
- ISBN-13: 978-1-57955-096-7 (paperback)
- ISBN-13: 978-1-57955-097-4 (Kindle)
- ISBN-13: 978-1-57955-095-0 (Wolfram Notebooks)
Contents
- Introduction
- What Is Complex Analysis?
- The Complex Plane
- Complex Functions
- The Exponential Function
- The Argument Function
- The Logarithm Function and Complex Powers
- Limits and Continuity
- The Point at Infinity
- Complex Derivatives
- The Cauchy–Riemann Equations
- Complex Line Integrals
- Fundamental Theorem for Complex Line Integrals
- Cauchy's Theorem
- Applications of Cauchy's Theorem
- Cauchy's Integral Formula
- Three Important Theorems
- Harmonic Functions
- Properties of Harmonic Functions
- Power Series
- Taylor Series
- Laurent Series
- Holomorphic and Meromorphic Functions
- Residues
- The Residue Theorem
- Transcendental Equations
- Definite Integrals
- Gamma Function
- Laplace Transforms
- Hydrodynamics
- Elliptic Functions
- Complex Analysis in a Nutshell
- Sample Final Exam
- References
Companion Online Course Introduction to Complex Analysis
Wolfram U interactive courses are hosted on the Wolfram Cloud and allow you to interactively explore concepts using Wolfram Language functionality. This interactive course includes video lessons, practice problems, quizzes, a final exam and a chat-based Course Assistant (requires AI Access). Two shareable certificates are available: course completion and Wolfram Level 1 certification for proficiency in complex analysis. Work through the course content at your own pace and use the built-in progress tracker as you advance. Get started with our free courses by signing in to the Wolfram Cloud with your Wolfram ID—or create one; it's free!
See Course Details