黎曼空間之積分運算

Abstract

在黎曼面上對封閉曲線的基底 a,b cycles 積分可以解決許多微分方程的問題。將複數平面推廣至黎曼面，使得一個定義在複數平面上的多值函數在黎曼面上是single-valued和analytic的。由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。同時，periodic soliton solution 也可由一簡單的方法獲得，並且能夠經由"Mathematica"得到驗證。

The integrals over a,b cycles for the cuts on Riemann surface will solve many problems in Differential Equations. Generalize the complex plane C to the Riemann surface such that one two-valued function defined on C becomes single-valued and analytic defined on Riemann surface. By Cauchy integral theorem, we find an equivalent path of a,b cycle such that two integrals equal. The equivalent path integrals along cuts can be computed by "Mathematica" simply and correctly. This approach offers an easy way to obtain the periodic soliton solution and be checked by "Mathematica".