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This book constitutes the first installment of a projected three-volume work that will present applications as well as the basic theory of analytic functions of one or several complex variables. Applications are made to other branches of mathematics, to science and engineering, and to computation. The algorithmic attitude toward mathematics—not to consider a problem solved unless an. algorithm for constructing the solution has been found—prevails not only in the sections devoted to computation but forms one of the work’s unifying themes. A short overview of the three volumes is in order. The first volume, after laying the necessary foundations in the theory of power series and of complex integration, discusses applications and basic theory (without the Riemann mapping theorem) of conformal mapping and the solution of algebraic and transcendental equations. The second volume will cover topics that are broadly connected with ordinary differential equations special functions, integral transforms, asymptotics, and continued fractions. The third volume will center similarly around partial differential equations and will feature harmonic functions, the construction of conformal maps, the Bergman-Vekua theory of elliptic partial differential equations with analytic coefficients, and analytical techniques for solving three-dimensional potential problems. In collecting all these topics under one cover, I have been guided by the idea that for today’s applied mathematician it is not enough to specialize, however deeply, in any single narrowly restricted area. He should also be made aware as forcefully as possible of the light that radiates from the basic theories of mathematics into the neighboring fields of science. What 1 have tried to do here for complex analysis, should as part of an applied mathematics curriculum also be done, for instance, in real analysis and linear algebra. A word about the layout of the contents of the present volume may be necessary. It is well known that there are two essentially different approaches to complex variable theory Riemann's approach, based on complex differentiability, and the approach of Weierstrass, bused on power series. Although the Weierstrassian example has been followed in a venerable series of classical texts (Whittaker and Watson [1927], first ed. [ I903, Hurwitz and Courant [1929], Dienes [1931]), the geometric method of Riemann, brilliantly exposed, for example, by Ahlfors [1953], seems to have been preferred by most analysts to the computational method of Weierstrass, at least until recently, when under the influence of Bourbaki (Cartan [1961]) power series came back into fashion.
formal Power Series.
Functions Analytic at a Point.
Analytic Continuation.
Complex Integration.
Conformal Mapping.
Polynomials.
Partial Fractions