Description

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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. Contents

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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 Related Topics

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Calculus and Analysis