Complex Analysis (3rd Edition) - Joseph Bak.pdf

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Undergraduate Texts in Mathematics
Editorial Board
S. Axler
K.A. Ribet
For other titles Published in this series, go to
http://www.springer.com/series/666
Joseph Bak • Donald J. Newman
Complex Analysis
Third Edition
1C
Joseph Bak
City College of New York
Department of Mathematics
138th St. & Convent Ave.
New York, New York 10031
USA
jbak@ccny.cuny.edu
Donald J. Newman
(1930–2007)
Editorial Board:
S. Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132
USA
axler@sfsu.edu
K. A. Ribet
Mathematics Department
University of California at Berkeley
Berkeley, CA 94720
USA
ribet@math.berkeley.edu
ISSN 0172-6056
ISBN 978-1-4419-7287-3
e-ISBN 978-1-4419-7288-0
DOI 10.1007/978-1-4419-7288-0
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2010932037
Mathematics Subject Classification (2010): 30-xx, 30-01, 30Exx
© Springer Science+Business Media, LLC 1991, 1997, 2010
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Preface to the Third Edition
Beginning with the first 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 in this edition, which include additions to ten
of the nineteen chapters, are intended to provide the additional insights that can be
obtained by seeing a little more of the “big picture”. This includes additional related
results and occasional generalizations that place the results in a slightly 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
roots of a general polynomial, we added a section on Newton’s Method, 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.
A series of new results relate to the mapping properties of analytic functions.
A revised 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
Schwarz Reflection Principle has been expanded to include reflection across analytic
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.
A single interesting application to number theory in the earlier editions has been
expanded into a new section (19.4) which includes four examples from additive
number theory, all united in their use of generating functions.
Perhaps the most significant changes in this edition revolve around the proof of
the prime number theorem. There are two new sections (17.3 and 18.2) on Dirichlet
series. With that background, a pivotal result on the Zeta function (18.10), which
seemed to “come out of the blue”, is now seen in the context of the analytic con-
tinuation of Dirichlet series. Finally the actual proof of the prime number theorem
has been considerably revised. The original independent proofs by Hadamard and
de la Vallée Poussin were both long and intricate. Donald Newman’s 1980 article
v

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