Applied and Computational Complex Analysis - Volume 2: Special Functions, Integral Transforms, Asymptotics, Continued Fractions (v. 2)

At a mathematical level accessible to the non-specialist, the third of a three-volume work shows how to use methods of complex analysis in applied mathematics and computation. The book examines two-dimensional potential theory and the construction of conformal maps for simply and multiply connected regions. In addition, it provides an introduction to the theory of Cauchy integrals and their applications in potential theory, and presents an elementary and self-contained account of de Branges' recently discovered proof of the Bieberbach conjecture in the theory of univalent functions. The proof offers some interesting applications of material that appeared in volumes 1 and 2 of this work. It discusses topics never before published in a text, such as numerical evaluation of Hilbert transform, symbolic integration to solve Poisson's equation, and osculation methods for numerical conformal mapping. Presents applications as well as the basic theory of analytic functions of one or several complex variables. The first volume discusses applications and basic theory of conformal mapping and the solution of algebraic and transcendental equations. Volume Two covers topics broadly connected with ordinary differental equations: special functions, integral transforms, asymptotics and continued fractions. Volume Three details discrete fourier analysis, cauchy integrals, construction of conformal maps, univalent functions, potential theory in the plane and polynomial expansions. Fourier analysis is one of the most pervasive tools in applied analysis.