黎曼空間之積分運算

Abstract

在複數平面上是多值和不可解析的函數。我們利用代數與幾何方法定義一個新曲面”黎曼面”替代複數平面，使得一個定義在複數平面上是多值的函數在黎曼面上是唯一值且可解析的函數。在黎曼面上對封閉曲線的基底a，b，c cycles積分可以解決許多微分方程的問題。我們可以找到 a，b，c cycles 的等價路徑，由Cauchy Integral Theorem得a，b，c cycles積分值與它們的等價路徑積分值相等。利用Mathematica，等價路徑的積分值可以被正確的求出。

is a m-valued function on complex plane C. We use algebraic and geometric analysis to develop a new surface, namely, the Riemann surfaces R such that f becomes single-valued and analytic on R. The integrals over a, b, c cycles on Riemann surface can solve many problems in Differential Equations. By Cauchy Integral Theorem, the integrals over a,b,c cycles on R are equivalent to the integrals over equivalent simple paths. The integrals of equivalent paths can be computed by "Mathematica" correctly.