Mathematical Sciences: Classical Complex Analysis

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Proposal: DMS-9800464 Principal Investigator: Donald E. Marshall Abstract: Under this grant Marshall will investigate problems in four areas of analytic function theory. Estimates of harmonic measure and the growth of hyperbolic distance will be used to study the angular distribution of mass induced by analytic, area-integrable functions. This problem has applications to the characterization of extremal dilatations for quasi-conformal homeomorphisms of the unit disk. Secondly he will investigate the integrability properties of derivatives of one-to-one analytic functions. Thirdly, he will investigate finite interpolation problems for the bidisk. Marshall will seek continued fraction decompositions for a certain class of rational functions in several variables to solve this problem. In the fourth area, Marshall will investigate the accuracy of a promising technique for the numerical computation of conformal maps. Conformal maps (one-to-one analytic functions) have been used as a tool in science and engineering for many years. One way conformal maps are used is to transform a problem on a complicated region in the complex plane to a related problem on a "standard" region, such as a disk or half-plane, where known techniques can be used. The solution on the standard region is then transformed by the inverse of the conformal map to a solution of the original problem on the original region. Classically, this method was used for problems related to Laplace's equation. For example, temperature at equilibrium on a thin metallic plate satisfies Laplace's equation. More recently, conformal maps have found application to a wide range of numerical solutions of other partial differential equations arising in engineering problems. There are applications in electro-magnetics, vibrating membranes and acoustics, transverse vibrations and buckling of plates, elasticity, heat transfer, and fluid flow, for example. While early applications used explicit analytic representations for conformal maps, modern us es require conformal maps of more complicated regions which cannot be represented easily in terms of elementary functions. The only resort is to compute numerical approximations to the desired conformal maps. We will investigate the accuracy of a new technique which rapidly computes conformal maps and their inverses. It is fast enough that it can be used for experimentation on a typical workstation. The integrability questions we will work on deal with the difficult problem of estimating the growth rate of conformal maps. One of the applications of complex analysis to electrical engineering problems is the construction of electric circuits associated with a given transfer function. One method is to decompose the transfer function into simpler pieces, called a continued fraction expansion. In order to understand related problems in several complex variables, Marshall will seek similar decompositions for transfer functions depending on two variables.