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Full Description
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.
Contents
The Complex Numbers.- Functions of the Complex Variable z.- Analytic Functions.- Line Integrals and Entire Functions.- Properties of Entire Functions.- Properties of Analytic Functions.- Further Properties of Analytic Functions.- Simply Connected Domains.- Isolated Singularities of an Analytic Function.- The Residue Theorem.- Applications of the Residue Theorem to the Evaluation of Integrals and Sums.- Further Contour Integral Techniques.- to Conformal Mapping.- The Riemann Mapping Theorem.- Maximum-Modulus Theorems for Unbounded Domains.- Harmonic Functions.- Different Forms of Analytic Functions.- Analytic Continuation; The Gamma and Zeta Functions.- Applications to Other Areas of Mathematics.
