Essentials of Complex Analysis A Computational Approach

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.

Information and Media Inquiries

Fall 2025 Publication
Publicity and Interviews: publishing@wolfram.com
Trim Size: 7" x 10"
Non-Fiction

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

About the Author

Marco Saragnese is a Kernel Developer at Wolfram Research. He holds a PhD in physics from the University of Hamburg. His research interest lies in the area of theoretical particle physics, particularly the computation of scattering amplitudes. At Wolfram, he has developed the functions for line, surface and complex contour integrals and is also involved in other symbolic integration projects.