| dc.description.abstract | 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’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.
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. | en_US |
