黎曼空間之積分運算

Abstract

黎曼面上對封閉曲線的基底 a,b cycles 積分可以解決部分微分方程的問題。將複數平面推廣至黎曼面，使得定義在複數平面上的多值函數在黎曼面上是單值和可解析的。我們利用代數及幾何方法來定義此種三面的黎曼空間,然後利用Cauchy integral theorem，我們可以找到一組與 a,b cycles 等價的路徑，使得兩種積分相等。利用Mathematica，此組沿著 cuts 的等價路徑之積分可以被正確且簡單的求出。

The integrals over a,b cycles on Riemann surface can solve some problems in differential Equations. We use algebraic and geometric method to construct three-sheeted Riemann surfaces of genus N such that a multi-valued function is single-valued and analytic on the Riemann surface. Then, by Cauchy integral theorem, we can find equivalent paths of a,b cycles such that the two integrals are equal. The equivalent path-integrals along cuts can be computed by "Mathematica" simply and correctly.