復變函數(shù)與積分變換-isee

1、Engineering Mathematics,Complex Variables & ApplicationsChapter 4,鄭偉詩wszheng@ieee.org, http://sist.sysu.edu.cn/~zhwshi/,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 2,Outlilne,1、Definition of Integral,2、C

2、ondition for Existence of Integral and Methods of Calculation,3、Properties of Integral,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 3,Curve, Contours,arc,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page

3、 4,Contours,When the arc C is simple except for the fact that z(b)=z(a), we say Cis a simple closed curve, or a Jordan curve.,Simple arc / Jordan arc,The arc C is a simple arc, or a Jordan arc, if it does not cross its

4、elf.,Simple closed curve / Jordan curve,The positive orientation is the counterclockwise direction.,Positively oriented curve,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 5,5,Contours,Contour,Differentiable arc,Le

5、ngth of C,Simple closed contour,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 6,6,Contour Integral,,,,,,,,,,,Suppose function is defined in domain D, C is a contour in D from point A to point B. Divid

6、e curve C into n segmented lines, the points of division are denoted by,Randomly pick a point from each segment of curve,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 7,7,(,If has an unique limit reg

7、ardless of the division of C and partition method of ,then we call this limit value as the integral of function on curve C, denoted by,Contour Integral,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 8,Co

8、ntour Integral,Along a contour C,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 9,Contour Integral,To compute,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 10,About the definition:,then this definition is same

9、 to the definition of integral for single real variable function.,Contour Integral,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 11,11,*Example1:,*Solution:,The line equation is,Contour Integral,Wei-Shi Zhengwszhen

10、g@ieee.org,2024/3/16, Page 12,12,,,these two integral have nothing do with path-integral C,then regardless of the curve movementto point,Contour Integral,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 13,13,*Exa

11、mple 2:,*Solution:,(1) The parametric equation is,,y=x,Contour Integral,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 14,(2) parametric equation is,,Contour Integral,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Pag

12、e 15,(3) integration path is composed by two line segments,,,parametric equation of straight-line segment along x-axis is,parametric equation of straight-line segment from point 1 to point 1+i is,Contour Integral,Wei-Shi

13、 Zhengwszheng@ieee.org,2024/3/16, Page 16,16,*Example 3:,*Solution:,Parametric equation of integration path,(since |z|=2),Contour Integral,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 17,17,*Example 4:,*Soluti

14、on:,Parametric equation of integrationpath is:,,,,,Contour Integral,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 18,18,,Important Conclusion: integral value is independent to the center point and radius of the ci

15、rcle.,when n=0,when,Contour Integral,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 19,With Branch Cut,Contour Integral,?,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 20,Properties of Integral,Complex integral

16、has similar properties with definite integral of real variable function.,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 21,Properties of Integral,板書證明,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 22,Anti-Deriva

17、tives,板書證明,,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 23,,,? Not D but a curve,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 24,Cauchy–Goursat theorem,,,板書證明,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, P

18、age 25,Cauchy–Goursat theorem,,Applications:,,simple closed contour, closed contours (intersection: finite / infinite),板書證明,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 26,Cauchy–Goursat theorem,Example:,Wei-Shi Z

19、hengwszheng@ieee.org,2024/3/16, Page 27,Recall the following theorem,Cauchy–Goursat theorem,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 28,Cauchy–Goursat theorem,板書證明,Wei-Shi Zhengwszheng@ieee.org,2024/3/16,

20、 Page 29,Cauchy–Goursat theorem,principle of deformation of paths,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 30,Cauchy–Goursat theorem,Example:,?,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 31,Cauchy I

21、ntegral Formula,板書證明,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 32,Cauchy Integral Formula,,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 33,Cauchy Integral Formula,Gauss's mean value theorem,,Wei-Shi

22、Zhengwszheng@ieee.org,2024/3/16, Page 34,Extensions: Analytic,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 35,Extensions: Analytic,,,,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 36,Extensions: Analyti

23、c,,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 37,Extension: Liouville’s theorem,,,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page 38,Extension: Max Modulus,,Wei-Shi Zhengwszheng@ieee.org,2024/3/16, Page