Complex Analysis

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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
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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.

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