1、Engineering Mathematics,Complex Variables & ApplicationsChapter 4,郑伟诗 wszhengieee.org, http:/ Zheng wszhengieee.org,2019/2/24, Page 2,Outlilne,1、Definition of Integral,2、Condition for Existence of Integral and Methods of Calculation,3、Properties of Integral,Wei-Shi Zheng wszhengieee.org,2019/2/24, P
2、age 3,Curve, Contours,arc,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 4,Contours,When the arc C is simple except for the fact that z(b)=z(a), we say C is 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 itself.,Simpl
3、e closed curve / Jordan curve,The positive orientation is the counterclockwise direction.,Positively oriented curve,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 5,5,Contours,Contour,Differentiable arc,Length of C,Simple closed contour,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 6,6,Contour Integr
4、al,Suppose function is defined in domain D, C is a contour in D from point A to point B. Divide curve C into n segmented lines, the points of division are denoted by,Randomly pick a point from each segment of curve,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 7,7,(,If has an unique limit regardless
5、 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 Zheng wszhengieee.org,2019/2/24, Page 8,Contour Integral,Along a contour C,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 9,Contour Integral,To comp
6、ute,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 10,About the definition:,then this definition is same to the definition of integral for single real variable function.,Contour Integral,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 11,11,*Example1:,*Solution:,The line equation is,Contour Integral,We
7、i-Shi Zheng wszhengieee.org,2019/2/24, Page 12,12,these two integral have nothing do with path-integral C,then regardless of the curve movement to point,Contour Integral,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 13,13,*Example 2:,*Solution:,(1) The parametric equation is,y=x,Contour Integral,Wei
8、-Shi Zheng wszhengieee.org,2019/2/24, Page 14,(2) parametric equation is,Contour Integral,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 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 f
9、rom point 1 to point 1+i is,Contour Integral,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 16,16,*Example 3:,*Solution:,Parametric equation of integration path,(since |z|=2),Contour Integral,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 17,17,*Example 4:,*Solution:,Parametric equation of integration
10、 path is:,Contour Integral,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 18,18,Important Conclusion: integral value is independent to the center point and radius of the circle.,when n=0,when,Contour Integral,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 19,With Branch Cut,Contour Integral,?,Wei-Shi
11、Zheng wszhengieee.org,2019/2/24, Page 20,Properties of Integral,Complex integral has similar properties with definite integral of real variable function.,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 21,Properties of Integral,板书证明,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 22,Anti-Derivatives,板书证
12、明,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 23,? Not D but a curve,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 24,CauchyGoursat theorem,板书证明,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 25,CauchyGoursat theorem,Applications:,simple closed contour, closed contours (intersection: finite / infin
13、ite),板书证明,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 26,CauchyGoursat theorem,Example:,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 27,Recall the following theorem,CauchyGoursat theorem,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 28,CauchyGoursat theorem,板书证明,Wei-Shi Zheng wszhengieee.org,2019
14、/2/24, Page 29,CauchyGoursat theorem,principle of deformation of paths,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 30,CauchyGoursat theorem,Example:,?,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 31,Cauchy Integral Formula,板书证明,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 32,Cauchy Integral Form
15、ula,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 33,Cauchy Integral Formula,Gausss mean value theorem,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 34,Extensions: Analytic,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 35,Extensions: Analytic,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 36,Extensio
16、ns: Analytic,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 37,Extension: Liouvilles theorem,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 38,Extension: Max Modulus,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 39,Extension: Max Modulus,Wei-Shi Zheng wszhengieee.org,2019/2/24, Page 40,Extension: Max Modulus,
