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The Calculus of Complex Functions
 
William Johnston Butler University, Indianapolis, IN
The Calculus of Complex Functions
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6565-0
Product Code:  TEXT/71
List Price: $85.00
MAA Member Price: $63.75
AMS Member Price: $63.75
eBook ISBN:  978-1-4704-6901-6
Product Code:  TEXT/71.E
List Price: $85.00
MAA Member Price: $63.75
AMS Member Price: $63.75
Softcover ISBN:  978-1-4704-6565-0
eBook: ISBN:  978-1-4704-6901-6
Product Code:  TEXT/71.B
List Price: $170.00 $127.50
MAA Member Price: $127.50 $95.63
AMS Member Price: $127.50 $95.63
The Calculus of Complex Functions
Click above image for expanded view
The Calculus of Complex Functions
William Johnston Butler University, Indianapolis, IN
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6565-0
Product Code:  TEXT/71
List Price: $85.00
MAA Member Price: $63.75
AMS Member Price: $63.75
eBook ISBN:  978-1-4704-6901-6
Product Code:  TEXT/71.E
List Price: $85.00
MAA Member Price: $63.75
AMS Member Price: $63.75
Softcover ISBN:  978-1-4704-6565-0
eBook ISBN:  978-1-4704-6901-6
Product Code:  TEXT/71.B
List Price: $170.00 $127.50
MAA Member Price: $127.50 $95.63
AMS Member Price: $127.50 $95.63
  • Book Details
     
     
    AMS/MAA Textbooks
    Volume: 712022; 433 pp
    MSC: Primary 30; Secondary 47

    The book introduces complex analysis as a natural extension of the calculus of real-valued functions. The mechanism for doing so is the extension theorem, which states that any real analytic function extends to an analytic function defined in a region of the complex plane. The connection to real functions and calculus is then natural. The introduction to analytic functions feels intuitive and their fundamental properties are covered quickly. As a result, the book allows a surprisingly large coverage of the classical analysis topics of analytic and meromorphic functions, harmonic functions, contour integrals and series representations, conformal maps, and the Dirichlet problem. It also introduces several more advanced notions, including the Riemann hypothesis and operator theory, in a manner accessible to undergraduates. The last chapter describes bounded linear operators on Hilbert and Banach spaces, including the spectral theory of compact operators, in a way that also provides an excellent review of important topics in linear algebra and provides a pathway to undergraduate research topics in analysis.

    The book allows flexible use in a single semester, full-year, or capstone course in complex analysis. Prerequisites can range from only multivariate calculus to a transition course or to linear algebra or real analysis. There are over one thousand exercises of a variety of types and levels. Every chapter contains an essay describing a part of the history of the subject and at least one connected collection of exercises that together comprise a project-level exploration.

    Ancillaries:

    Readership

    Undergraduate students interested in analysis.

  • Table of Contents
     
     
    • Cover
    • Title page
    • Copyright
    • Contents
    • Preface
    • Chapter 1. Analytic Functions and the Derivative
    • 1.1. The Complex Derivative
    • 1.2. Power Series and the Extension Theorem
    • 1.3. Multivalued Functions and Riemann* Surfaces
    • 1.4. The Cauchy-Riemann Equations and* Harmonic Functions
    • 1.5. Analytic Continuation
    • Notes for Chapter 1
    • Chapter 2. Complex Integration
    • 2.1. Complex Line Integrals
    • 2.2. The Cauchy-Goursat Theorem
    • 2.3. Cauchy’s Integral Theorem
    • 2.4. Top Ten Facts from Cauchy’s Integral* Theorem
    • Notes for Chapter 2
    • Chapter 3. Non-Entire Functions
    • 3.1. Singularities
    • 3.2. Laurent Series
    • 3.3. Cauchy’s Residue Theorem and the* Argument Principle
    • 3.4. Applications of the Residue Theorem
    • Notes for Chapter 3
    • Chapter 4. Solving the Dirichlet Problem
    • 4.1. Conformal Maps
    • 4.2. Solutions via Streamlines and* Equipotentials
    • 4.3. Solutions via Green’s Functions
    • 4.4. Solutions via an Integral Representation
    • Notes for Chapter 4
    • Chapter 5. Further Topics and Famous Discoveries
    • 5.1. Integral Transforms
    • 5.2. Analytic Number Theory
    • 5.3. The Riemann Hypothesis
    • 5.4. Generalizing the Fundamental Theorem of Algebra
    • Notes for Chapter 5
    • Chapter 6. Linear Algebra and Operator Theory
    • 6.1. Bounded Linear Operators on a Hilbert* Space
    • 6.2. The Study of Finite-Rank Operators: Linear Algebra
    • 6.3. Banach Spaces and Compact Operators
    • 6.4. Open Research Questions
    • Notes for Chapter 6
    • Acknowledgments and Credits
    • Solutions to Odd Exercises
    • Bibliography
    • Index
    • Back Cover
  • Reviews
     
     
    • This is an excellent self-explanatory and self-contained textbook for senior undergraduate or freshman graduate complex analysis and number theory students. It clearly explains how theoretical complex analysis can help mathematicians answer difficult questions about real numbers, in particular, questions about integers. Each of the six chapters starts with an easy explanation of the topic and then brilliantly moves to more involved problems and discoveries. Each section comes with both helpful solved examples as well as a comprehensive set of exercises with answers to most problems. These topics and their related exercises are beautifully illustrated using Mathematica graphing technology. What is special about this textbook is that at the end of each chapter, a history of the leading mathematicians and their discoveries is nicely narrated, which can be good motivation for curious young mathematicians to further explore and investigate these topics. These short, but informative, biographies explain how our current understanding of complex analysis evolves from three distinct lines of development, arising from the work of Riemann, Cauchy, and Weierstrass.

      Jay M. Jahangiri (Kent State University), MathSciNet
    • In addition to being well organized, this is a well written and beautifully illustrated text. ...The narrative voice is precise and pays careful attention to the analytic details, but remains somewhat informal and steers away from being too dry. There are a number of features that embellish the text, including embedded Explorations, end-of-section Important Concepts, and end-of-chapter Historical Notes. All of these are appreciated — the historical notes are a particularly nice touch. Scattered throughout the text are Engaged Learning Modules, which offer explorations in Mathematica or in external sites and applets. Each section concludes with an assortment of exercises that seem appropriately challenging.

      John Ross, Southwestern University
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 712022; 433 pp
MSC: Primary 30; Secondary 47

The book introduces complex analysis as a natural extension of the calculus of real-valued functions. The mechanism for doing so is the extension theorem, which states that any real analytic function extends to an analytic function defined in a region of the complex plane. The connection to real functions and calculus is then natural. The introduction to analytic functions feels intuitive and their fundamental properties are covered quickly. As a result, the book allows a surprisingly large coverage of the classical analysis topics of analytic and meromorphic functions, harmonic functions, contour integrals and series representations, conformal maps, and the Dirichlet problem. It also introduces several more advanced notions, including the Riemann hypothesis and operator theory, in a manner accessible to undergraduates. The last chapter describes bounded linear operators on Hilbert and Banach spaces, including the spectral theory of compact operators, in a way that also provides an excellent review of important topics in linear algebra and provides a pathway to undergraduate research topics in analysis.

The book allows flexible use in a single semester, full-year, or capstone course in complex analysis. Prerequisites can range from only multivariate calculus to a transition course or to linear algebra or real analysis. There are over one thousand exercises of a variety of types and levels. Every chapter contains an essay describing a part of the history of the subject and at least one connected collection of exercises that together comprise a project-level exploration.

Ancillaries:

Readership

Undergraduate students interested in analysis.

  • Cover
  • Title page
  • Copyright
  • Contents
  • Preface
  • Chapter 1. Analytic Functions and the Derivative
  • 1.1. The Complex Derivative
  • 1.2. Power Series and the Extension Theorem
  • 1.3. Multivalued Functions and Riemann* Surfaces
  • 1.4. The Cauchy-Riemann Equations and* Harmonic Functions
  • 1.5. Analytic Continuation
  • Notes for Chapter 1
  • Chapter 2. Complex Integration
  • 2.1. Complex Line Integrals
  • 2.2. The Cauchy-Goursat Theorem
  • 2.3. Cauchy’s Integral Theorem
  • 2.4. Top Ten Facts from Cauchy’s Integral* Theorem
  • Notes for Chapter 2
  • Chapter 3. Non-Entire Functions
  • 3.1. Singularities
  • 3.2. Laurent Series
  • 3.3. Cauchy’s Residue Theorem and the* Argument Principle
  • 3.4. Applications of the Residue Theorem
  • Notes for Chapter 3
  • Chapter 4. Solving the Dirichlet Problem
  • 4.1. Conformal Maps
  • 4.2. Solutions via Streamlines and* Equipotentials
  • 4.3. Solutions via Green’s Functions
  • 4.4. Solutions via an Integral Representation
  • Notes for Chapter 4
  • Chapter 5. Further Topics and Famous Discoveries
  • 5.1. Integral Transforms
  • 5.2. Analytic Number Theory
  • 5.3. The Riemann Hypothesis
  • 5.4. Generalizing the Fundamental Theorem of Algebra
  • Notes for Chapter 5
  • Chapter 6. Linear Algebra and Operator Theory
  • 6.1. Bounded Linear Operators on a Hilbert* Space
  • 6.2. The Study of Finite-Rank Operators: Linear Algebra
  • 6.3. Banach Spaces and Compact Operators
  • 6.4. Open Research Questions
  • Notes for Chapter 6
  • Acknowledgments and Credits
  • Solutions to Odd Exercises
  • Bibliography
  • Index
  • Back Cover
  • This is an excellent self-explanatory and self-contained textbook for senior undergraduate or freshman graduate complex analysis and number theory students. It clearly explains how theoretical complex analysis can help mathematicians answer difficult questions about real numbers, in particular, questions about integers. Each of the six chapters starts with an easy explanation of the topic and then brilliantly moves to more involved problems and discoveries. Each section comes with both helpful solved examples as well as a comprehensive set of exercises with answers to most problems. These topics and their related exercises are beautifully illustrated using Mathematica graphing technology. What is special about this textbook is that at the end of each chapter, a history of the leading mathematicians and their discoveries is nicely narrated, which can be good motivation for curious young mathematicians to further explore and investigate these topics. These short, but informative, biographies explain how our current understanding of complex analysis evolves from three distinct lines of development, arising from the work of Riemann, Cauchy, and Weierstrass.

    Jay M. Jahangiri (Kent State University), MathSciNet
  • In addition to being well organized, this is a well written and beautifully illustrated text. ...The narrative voice is precise and pays careful attention to the analytic details, but remains somewhat informal and steers away from being too dry. There are a number of features that embellish the text, including embedded Explorations, end-of-section Important Concepts, and end-of-chapter Historical Notes. All of these are appreciated — the historical notes are a particularly nice touch. Scattered throughout the text are Engaged Learning Modules, which offer explorations in Mathematica or in external sites and applets. Each section concludes with an assortment of exercises that seem appropriately challenging.

    John Ross, Southwestern University
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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