数学物理方法Chapt 1 _3.pptxVIP

按时交作业（周一留的作业，下周一交） 多次晚交作业会影响平时成绩（偶尔的特殊原因除外） 作业本（作业纸）学号请写8位 ，如1121xxxx， 不要只写后4位。因为同时有12级和11级。 上一堂课内容 极限不同趋近路径：讲的都是自变量的趋近路径 实数，沿实轴趋近； 复数，在复平面里趋近。;Chapter 1: Complex functions 复函数;Contour Integrals 围线积分;Chapt 1: Complex functions;An alternative definition of counter integral, which reduces the complex integral to the complex sum of real integrals.;?;In everyday language, a simply connected region is one that has no holes.;Statement of Theorem;Examples that do (and do not) meet Cauchy’s integral theorem;Discussions about Example 1;Chapt 1: Complex functions;Cauchy’s Theorem: Proof;This establishes the theorem;了解的内容;Multiply Connected Regions;Cauchy’s integral theorem is not valid for the contour C, but we can construct a contour C’ for which the theorem holds.;Interpretation to What we have shown is that the integral of an analytic function over a closed contour surrounding an “island” of nonanalyticity can be subjected to any continuous deformation within the region of analyticity without changing the value of the integral. The notion of continuous deformation means that the change in contour must be able to be carried out via a series of small steps, which precludes processes whereby we “jump over” a point or region of nonanalyticity. ;Looking back at the two examples of this section we see that the integrals of z2 (or z) vanished for both the circular and square contours, as prescribed by Cauchy’s integral theorem for an analytic function. The integrals of z-1 did not vanish, and vanishing was not required because there was a point of nonanalyticity within the contours. the integrals of z-1 for the two contours had the same value, as either contour can be reached by continuous deformation of the other.;extension to Example 1 plus the fact that closed contours in a region of analyticity can be deformed continuously without altering the value of the integral, we have the valuable and useful result: 下次课讲 Cauchy’s integral formula