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標題: The conformal mapping on a disk
作者: 阮奕彰
Yi-Chang Juan
關鍵字: 保角
Poisson integral
harmonic conjugate
引用: James Ward Brown and Ruel V. Churchill, Complex variables and applications 8th edition, Mc Graw Hill, 2009. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Springer Verlag, 1983. Xianfeng David Gu and Shing-Tung Yau, Computational conformal geometry, Higher Education Press, 2008. Lok Ming Lui, Ka Chun Lan, Shin-Tung Yau and Xianfeng Gu, Teichmuller extremal mapping and its applications to landmark matching registration, arXiv:1211.2569v1 [cs.CG], 2012. Walter A. Strauss, Parttial differential equations an introduction, John Wiley & Sons, Inc. 1992.
摘要: In this thesis, we employ the Poisson integral formula to derive a harmonic function u on a disc, and after that we create its conjugate function v . The complex function f(x,y)=u(x,y)+iv(x,y) is then an analytic function on a disk. Numerically we show that the function f is a conformal mapping. The image D_2 of the function f from a unit disk D_1 may not be a unit disk. We leave this interesting problem of finding a homeomorphism mapping from D_2 onto a unit disk for further study. Finally, we will adopt numerical experiments to demonstrate the feasibility of our theory. This paper is the first use of the Poisson integral formula to produce conformal function, there still are many interesting subjects for further mathematical improving. We provide a new area of research. Since the domain is a unit disk, it is reasonable to consider polar Laplace equation, which is riangle u = u_{rr} + {1over r} , u_r + {1over r^2} , u_{ heta heta}. For this equation, the origin point is a not defined. Therefore, it is challenged to solve the problem at the origin. We leave this problem for our further research.
在本篇論文中,我們利用了Poisson積分公式,製造圓盤上的調和函數u,然後求其調和共軛函數v ,則函數f(x,y)=u(x,y)+iv(x,y) 是定義在複數平面圓盤上的解析函數。然後利用電腦數值計算檢驗此函數具有保角性質。最後我們使用數值實驗,來製圖描述單位圓盤經保角映射後的值域。之後的研究將著重在如何找一個映射將保角映射後的值域再一對一且映成地送到圓盤的探討。此論文是首次使用Poisson積分公式來製造保角函數,尚有多處需要更進一步的數學印證。我們也因此提供一個新的研究領域。Laplace 方程式 riangle u=0 的解是一個調和函數。 但當定義是在單位圓盤作數值計算時,Laplace 方程式轉為極座標方程式較方便,即 riangle u = u_{rr} + {1over r} , u_r + {1over r^2} , u_{ heta heta} 但其缺點是此方程式在原點沒有定義。因此對於原點找出較佳的計算方法是俱有挑戰性的問題。我們預留此問題作為未來的研究。
文章公開時間: 2017-08-31
Appears in Collections:應用數學系所



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