论文标题
通过WeierStrass函数解决共形映射问题的解决方案
On the solution of a conformal mapping problem by means of Weierstrass functions
论文作者
论文摘要
考虑到矩形大坝下的多孔材料填充到上半平面的通道部分的共形映射问题。液压结构中流体流量的计算也出现了类似的问题。作为解决方案方法，使用了基督佛尔 - 距离 - 什瓦茨椭圆形积分在Weierstrass函数方面的表示。该计算基于Sigma函数的Taylor序列，其系数是递归确定的。获得了共形映射的简单公式，该公式取决于四个参数并使用Sigma函数。针对特定区域进行了数值实验。该区域的变性由趋于零的大坝宽度组成，并被考虑，结果表明，所得公式具有实现限制问题解决方案的极限。提出了Sigma功能泰勒系列系列系数的Weierstrass递归公式的精致证明。
The conformal mapping problem for the section of a channel filled with porous material under a rectangular dam onto the upper half-plane is considered. Similar problems arise in computing of fluid flow in hydraulic structures. As a solution method, the representation of Christoffel-Schwartz elliptic integral in terms of Weierstrass functions is used. The calculation is based on Taylor series for the sigma function, the coefficients of which are determined recursively. A simple formula for a conformal mapping is obtained, which depends on four parameters and uses the sigma function. A numerical experiment was carried out for a specific area. The degeneration of the region, which consists in the dam width tending to zero, is considered, and it is shown that the resulting formula has a limit that implements the solution of the limiting problem. A refined proof of Weierstrass recursive formula for the coefficients of Taylor series of the sigma function is presented.
