Title: 黎曼空間之積分運算II
Integral Evaluations on Three-sheeted Riemann Surfaces of Genus N of Type II
Authors: 蕭宇蔚
Hsiao, Yu-Wei
Lee, Jong-Eao
Keywords: 黎曼空間;黎曼曲面;開立方積分;分歧點;分歧割線;封閉曲線積分;多值函數;Riemann;Riemann Surface;three_sheeted Riemann Surface;branch point;branch cut;closed-curve integrals;multi-valued function
Issue Date: 2000
Abstract: 我們利用代數與幾何分析的方法建構多值函數(開立方函數)的黎曼曲面使其在此黎曼曲面上是單值函數而且是可解析的。在黎曼曲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,並由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,再利用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 solution 也可由此簡單的方法獲得,並且經由Mathematica得到驗證。
We use algebraic and geometric analysis to develop three-sheeted Riemann surface 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 find equivalent paths of a,b cycles such that their integrals are equal. And we use the software Mathematica to compute the equivalent paths of a,b cycles simply and correctly. This approach offers an easy way to obtain the periodic solution and be checked by Mathematica.
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