# Complex Analysis

Springer Science & Business Media, 02.08.2010 - 328 Seiten
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.

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### Inhalt

 The Complex Numbers 1 Functions of the Complex Variable z 21 Analytic Functions 35 Line Integrals and Entire Functions 44 Properties of Entire Functions 59 Properties of Analytic Functions 76 Further Properties of Analytic Functions 93 Simply Connected Domains 106
 Introduction to Conformal Mapping 169 The Riemann Mapping Theorem 195 MaximumModulus Theorems for Unbounded Domains 215 Chapter 16Harmonic Functions 225 Different Forms of Analytic Functions 240 Analytic Continuation The Gamma and Zeta Functions 257 Applications to Other Areas of Mathematics 273 Answers 291

 Isolated Singularities of an Analytic Function 117 The Residue Theorem 129 Applications of the Residue Theorem to the Evaluation of Integrals and Sums 143 Further Contour Integral Techniques 161
 References 319 Appendices 321 Index 325 Urheberrecht

### Über den Autor (2010)

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.