环上Szego核的无穷乘积表示
Infinite Product Representation for the Szego Kernel for an Annulus
摘要：
Szego核在共形映射问题中有许多应用，并且满足Kerzman-Stein积分方程。环的Szego核可以表示为双边级数。在本文中，我们展示了如何表示的Szego核作为一个基本的双边系列和无穷乘积表示通过应用的Ramanujan的总和。无穷乘积清楚地表明了Szego核的零点。并指出了它与基本Γ函数和修正雅可比θ函数的关系。将所得结果推广到一般环的Szego核和加权Szego核。本文还对基于双边级数和基于无穷乘积的Szego核的计算进行了数值比较。
英文摘要：
The Szego kernel has many applications to problems in conformal mapping and satisfies the Kerzman-Stein integral equation. The Szego kernel for an annulus can be expressed as a bilateral series. In this paper, we show how to represent the Szego kernel for an annulus as a basic bilateral series and an infinite product representation through the application of the Ramanujan's sum. The infinite product clearly exhibits the zero of the Szego kernel for an annulus. Its connection with basic gamma function and modified Jacobi theta function is also presented. The results are extended to the Szego kernel for general annulus and weighted Szego kernel. Numerical comparisons on computing the Szego kernel for an annulus based on the bilateral series and the infinite product are also presented.
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