N相黎曼空間的理論與應用

Abstract

我們利用代數與幾何分析的方法建構多值函數(開方函數)的黎曼空間使得一個定義在複數平面上是多值的函數在黎曼空間上是唯一值且可解析的函數。 在黎曼空間上對封閉曲線a，b cycles 的積分可以解決許多微分方程上的問題，而且可以找到 a，b cycles 之等價路徑，再經由Cauchy Integral Theorem可得知a，b， cycles 之積分值與它們的等價路徑積分值會相等。藉由這樣的方法，當我們執行黎曼空間的積分時，無論是數值上或是理論上，我們都可以解決問題進而求得解答。

We use algebraic and geometric analysis to develop two-sheet Riemannsurface R of genus N such that muti-valued function on the complex plane C become single-valued and analytic on R. The integrals over a,b cycles on R can solve many problems in Differential Equations. By Cauchy Integral Theorem, we can find equivalent paths of a,b cycles such that their integrals are equal. When we do the integral on the Riemann surface ,no matter what on theoretically or in value , by the principle ,we could solves the problem and get the solution.