Complex Analysis

1. Front Matter

   Pages i-xii

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2. The Complex Numbers

   • Joseph Bak, Donald J. Newman

   Pages 1-20

3. Functions of the Complex Variable z

   • Joseph Bak, Donald J. Newman

   Pages 21-34

4. Analytic Functions

   • Joseph Bak, Donald J. Newman

   Pages 35-43

5. Line Integrals and Entire Functions

   • Joseph Bak, Donald J. Newman

   Pages 45-57

6. Properties of Entire Functions

   • Joseph Bak, Donald J. Newman

   Pages 59-75

7. Properties of Analytic Functions

   • Joseph Bak, Donald J. Newman

   Pages 77-91

8. Further Properties of Analytic Functions

   • Joseph Bak, Donald J. Newman

   Pages 93-105

9. Simply Connected Domains

   • Joseph Bak, Donald J. Newman

   Pages 107-116

10. Isolated Singularities of an Analytic Function

    • Joseph Bak, Donald J. Newman

    Pages 117-128

11. The Residue Theorem

    • Joseph Bak, Donald J. Newman

    Pages 129-142

12. Applications of the Residue Theorem to the Evaluation of Integrals and Sums

    • Joseph Bak, Donald J. Newman

    Pages 143-160

13. Further Contour Integral Techniques

    • Joseph Bak, Donald J. Newman

    Pages 161-168

14. Introduction to Conformal Mapping

    • Joseph Bak, Donald J. Newman

    Pages 169-194

15. The Riemann Mapping Theorem

    • Joseph Bak, Donald J. Newman

    Pages 195-214

16. Maximum-Modulus Theorems for Unbounded Domains

    • Joseph Bak, Donald J. Newman

    Pages 215-223

17. Harmonic Functions

    • Joseph Bak, Donald J. Newman

    Pages 225-239

18. Different Forms of Analytic Functions

    • Joseph Bak, Donald J. Newman

    Pages 241-256

19. Analytic Continuation; The Gamma and Zeta Functions

    • Joseph Bak, Donald J. Newman

    Pages 257-272

20. Applications to Other Areas of Mathematics

    • Joseph Bak, Donald J. Newman

    Pages 273-290

Beginning with the ?rst edition of Complex Analysis, we have attempted to present the classical and beautiful theory of complex variables in the clearest and most intuitive form possible. The changes inthisedition, which include additions to ten of the nineteen chapters, are intended to provide the additional insights that can be obtainedby seeing a little more of the “bigpicture”.This includesadditional related results and occasional generalizations that place the results inaslightly broader context. The Fundamental Theorem of Algebra is enhanced by three related results. Section 1.3 offers a detailed look at the solution of the cubic equation and its role in the acceptance of complex numbers. While there is no formula for determining the rootsof a generalpolynomial,we added a section on Newton’sMethod,a numerical technique for approximating the zeroes of any polynomial. And the Gauss-Lucas Theorem provides an insight into the location of the zeroes of a polynomial and those of its derivative. Aseries of new results relate to the mapping properties of analytic functions. Arevised proof of Theorem 6.15 leads naturally to a discussion of the connection between critical points and saddle points in the complex plane. The proof of the SchwarzRe?ectionPrinciplehasbeenexpandedtoincludere?ectionacrossanalytic arcs, which plays a key role in a new section (14.3) on the mapping properties of analytic functions on closed domains. And our treatment of special mappings has been enhanced by the inclusion of Schwarz-Christoffel transformations.

• Analysis
• Complex analysis
• Funktionentheorie
• Residue theorem
• analytic function
• calculus
• maximum

From the reviews of the third edition:

“The book of the known mathematicians J. Bak and D. Newman is an excellent introduction into the theory of analytic functions of one complex variable. The book is written on an elementary level and so it supports students in the early stages of their mathematical studies. … The book also contains many illustrations, examples and exercises, which give additional information and explanations.” (Konstantin Malyutin, Zentralblatt MATH, Vol. 1205, 2011)

• Department of Mathematics, City College of New York, New York, USA

  Joseph Bak

Dr. Joseph Bak is the Assistant Chair of the Mathematics department at The City College of New York. Joseph Bak's primary area of research is approximation theory. Dr. Donald J. Newman (July 27, 1930 - March 28, 2007) was a champion problem solver. His mathematical specialties included complex analysis, approximation theory and number theory. His career included posts as a Professor of Mathematics at MIT, Brown University, Yeshiva University, Temple University and a distinguished chair at Bar Ilan University in Israel. His publications include 150 papers and five books.

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