Mathematical Sciences: Classical Complex Analysis

提案：DMS-9800464 首席研究员：Donald E. Marshall 摘要：在这笔资助下，Marshall 将研究解析函数理论四个领域的问题。调和测量的估计和双曲距离的增长将用于研究由解析面积可积函数引起的质量角分布。该问题可应用于单位圆盘准共形同胚的极值膨胀表征。其次，他将研究一对一解析函数的导数的可积性质。第三，他将研究 bidisk 的有限插值问题。马歇尔将在几个变量中寻求一类有理函数的连续分数分解来解决这个问题。在第四个领域，马歇尔将研究一种有前景的共形图数值计算技术的准确性。 共形图（一对一分析函数）多年来一直被用作科学和工程中的工具。使用共形图的一种方式是将复杂平面中的复杂区域上的问题变换为“标准”区域（例如圆盘或半平面）上的相关问题，其中可以使用已知技术。然后通过共形映射的逆将标准区域上的解变换为原始区域上原始问题的解。传统上，该方法用于解决与拉普拉斯方程相关的问题。例如，薄金属板上的平衡温度满足拉普拉斯方程。最近，共形映射已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，其在电磁学、振动膜和声学、板的横向振动和屈曲、弹性、传热和流体流动等领域都有应用。虽然早期应用使用共形映射的显式解析表示，但现代应用需要更复杂区域的共形映射，这些区域不能轻易地用初等函数表示。唯一的方法是计算所需共形图的数值近似值。 我们将研究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，可以用于在典型工作站上进行实验。我们将研究的可积性问题涉及估计共形图增长率的难题。复杂分析在电气工程问题中的应用之一是构建与给定传递函数相关的电路。一种方法是将传递函数分解为更简单的部分，称为连分式展开。为了理解几个复杂变量中的相关问题，马歇尔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.