Mathematical Sciences: Classical Complex Analysis

摘要：在此资助下，Marshall将研究解析函数理论的四个领域的问题。谐波测度的估计和双曲距离的增长将用于研究由解析的、面积可积的函数引起的质量的角分布。这个问题应用于单位盘的拟共形同胚的极值膨胀的刻画。其次，他将研究一对一解析函数导数的可积性。第三，他将研究双盘的有限插值问题。Marshall将寻求对某类多变量有理函数的连分式分解来解决这个问题。在第四个领域，Marshall将研究一种有前途的共形映射数值计算技术的准确性。保角映射（一对一解析函数）作为一种工具在科学和工程中已经使用了很多年。使用保角映射的一种方法是将复杂平面上的复杂区域上的问题转换为“标准”区域上的相关问题，例如可以使用已知技术的磁盘或半平面。然后将标准区域上的解通过保角映射的逆变换为原始问题在原始区域上的解。经典地，这种方法被用来解决与拉普拉斯方程有关的问题。例如，金属薄板上的平衡温度满足拉普拉斯方程。最近，保角映射已被广泛应用于工程问题中出现的其他偏微分方程的数值解。例如，在电磁学、振动膜和声学、横向振动和板的屈曲、弹性、传热和流体流动等方面都有应用。虽然早期的应用使用显式解析表示保角映射，但现代的应用需要更复杂的区域的保角映射，这些区域不容易用初等函数表示。唯一的办法是计算期望的保角映射的数值近似。我们将研究一种快速计算共形映射及其逆的新技术的准确性。它的速度足够快，可以在一个典型的工作站上进行实验。我们将研究的可积性问题涉及到估计共形映射增长率的难题。复杂分析在电气工程问题中的应用之一是与给定传递函数相关的电路的构造。一种方法是将传递函数分解成更简单的部分，称为连分式展开。为了理解几个复杂变量中的相关问题，Marshall将寻求依赖于两个变量的传递函数的类似分解。

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.