Mathematical Sciences: Classical Complex Analysis

摘要提案：DMS-9532087 PI：马歇尔 根据这项补助金，马歇尔将调查分析函数理论的三个领域的问题。 一种新的方法来构建良好的度量下限极值距离将适用于角导数问题的阿尔福斯1930年。 其次，他将研究一个有前途的新技术的数值计算的保角映射的准确性。第三，他将努力改善 Dirichlet空间及其乘子的插值函数的构造，使用再生核的更自然的线性组合。 保角映射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式是将复平面中的复杂区域上的问题转换为“标准”区域（诸如圆盘或半平面）上的相关问题，其中可以使用已知技术。 标准区域上的解然后通过保角映射的逆变换为原始区域上的原始问题的解。经典上，这种方法用于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。在过去的25年里，共形映射有许多非经典的应用，其中许多并不受拉普拉斯方程的约束。例如，在电磁学、振动膜和声学、板的横向振动和屈曲、弹性、传热和流体流动中有应用。 虽然早期的应用程序使用显式解析表示的共形映射，现代使用需要的共形映射更复杂的地区，不能很容易地表示在初等函数。因此，人们必须求助于数值逼近所需的共形映射。 马歇尔将研究一种新技术的准确性，这种技术可以快速计算共形映射及其逆。 这种技术足够快，可以在典型的工作站上进行实验。角导数的问题，他将调查是关于几何条件的边界上的一个区域，保证某些共形映射（定义内的区域）扩展到共形的一点上的边界。 圆盘上的狄利克雷函数不是共形的，但圆盘的图像具有有限的面积（计数重叠）。它们产生于总能量有界的势。马歇尔将研究的插值问题与在薄金属板上构建温度分布密切相关，该温度分布在板上的某些点处具有预先指定的值，并且具有最小的总能量。

ABSTRACT Proposal: DMS-9532087 PI: Marshall Under this grant, Marshall will investigate problems in three areas of analytic function theory. A new approach to constructing good metrics for lower bounds for extremal distance will be applied to the Angular Derivative Problem of Ahlfors 1930 . Secondly, he will investigate the accuracy of a promising new technique for the numerical computation of conformal maps. Thirdly, he will attempt to improve constructions of interpolating functions for the Dirichlet space and its multipliers, using more natural linear combinations of reproducing kernels. Conformal mapping has 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. There have been numerous non-classical applications of conformal maps developed in the last twenty five years, many of which are not governed by Laplace's equation. 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 uses require conformal maps of more complicated regions which cannot be represented easily in terms of elementary functions. One must therefore resort to numerical approximations to the desired conformal maps. Marshall will investigate the accuracy of a new technique which rapidly computes conformal maps and their inverses. This technique is fast enough that it can be used for experimentation on a typical workstation. The angular derivative problem he will investigate is about geometrical conditions on the boundary of a region that guarantee that certain conformal maps (defined inside the region) extend to be conformal at a point on the boundary. Dirichlet functions on the disk are not conformal, but the image of the disk has finite area (counting overlaps). They arise from potentials with bounded total energy. The interpolation problems Marshall will study are closely related to constructing a temperature distribution on a thin metal plate with preassigned values at certain points on the plate, and with smallest possible total energy.