Functions of a Complex Variable provides all the material for a
course on the theory of functions of a complex variable at the
senior undergraduate and beginning graduate level. Also suitable
for self-study, the book covers every topic essential to training
students in complex analysis. It also incorporates special topics
to enhance students' understanding of the subject, laying the
foundation for future studies in analysis, linear algebra,
numerical analysis, geometry, number theory, physics,
thermodynamics, or electrical engineering. After introducing the
basic concepts of complex numbers and their geometrical
representation, the text describes analytic functions, power series
and elementary functions, the conformal representation of an
analytic function, special transformations, and complex
integration. It next discusses zeros of an analytic function,
classification of singularities, and singularity at the point of
infinity; residue theory, principle of argument, Rouche's theorem,
and the location of zeros of complex polynomial equations; and
calculus of residues, emphasizing the techniques of definite
integrals by contour integration. The authors then explain uniform
convergence of sequences and series involving Parseval, Schwarz,
and Poisson formulas. They also present harmonic functions and
mappings, inverse mappings, and univalent functions as well as
analytic continuation.
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