Book Review:
Book Description
This is a textbook for an introductory course in complex analysis. It has been used for undergraduate complex analysis course at Georgia Institute of Technology.
This course is destined to introduce the student to the basic results in complex variable theory, in particular Cauchy's theorem, and to develop the student's facility in the following three areas:
 Computing the Laurent or Taylor series expansions associated to a function which is analytic in part of the complex plane, and the determination of the region of convergence of such series  Computing definite integrals by means of the residue calculus  Solving boundary value problems associated to the Laplace operator by means of the conformal transformations associated to analytic functions
Topics to be covered:
 Geometry of the complex plane, triangle inequalities, geometric proof of the fundamental theorem of algebra  Analytic functions: Continuity and differentiability, the CauchyRiemann equations. Complex functions as maps of the complex plane into itself  Elementary analytic functions, including the logarithm, and its principle branch, log(z)  Line integrals, the Cauchy integral formula and the CauchyGoursat theorem (proof of the Cauchy formula to be based on Green's theorem), Morera's theorem, etc.  Series: Taylor and Laurent expansions  Residue calculus for definite integrals  Harmonic functions and analytic functions: conjugate harmonics, conformal invariance of break Laplace's equation in the plane, the CauchyRiemann equations and conformal maps  Poisson kernel derived from Cauchy's formula; solution of boundary value problems for Laplace's equation by conformal mapping, selected applications
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