Engineering Mathematics Complex Variables & Applications Chapter 4 郑伟诗 wszheng@ieee.org, http://sist.sysu.edu.cn/~zhwshi/

Outlilne 1、Definition of Integral 2、Condition for Existence of Integral and Methods of Calculation 3、Properties of Integral Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 2

Curve, Contours arc Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 3

Contours Simple arc / Jordan arc Simple closed curve / Jordan curve The arc C is a simple arc, or a Jordan arc, if it does not cross itself. Simple closed curve / Jordan curve 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. Positively oriented curve The positive orientation is the counterclockwise direction. Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 4

Contours Differentiable arc Length of C Contour Simple closed contour Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 5 5

Contour Integral 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 wszheng@ieee.org 2019/7/8, Page 6 6

Contour Integral ( If has an unique limit regardless 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 Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 7 7

Contour Integral Along a contour C 2019/7/8, Page 8 Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 8

Contour Integral To compute 2019/7/8, Page 9 Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 9

Contour Integral About the definition: then this definition is same to the definition of integral for single real variable function. Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 10

Contour Integral *Example1: *Solution: The line equation is 11 Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 11 11

Contour Integral these two integral have nothing do with path-integral C then regardless of the curve movement to point Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 12 12

Contour Integral *Example 2: *Solution: (1) The parametric equation is y=x Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 13 13

Contour Integral (2) parametric equation is y=x 2019/7/8, Page 14 Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 14

Contour Integral (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 y=x Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 15

Contour Integral *Example 3: *Solution: Parametric equation of integration path (since |z|=2) Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 16 16

Contour Integral *Example 4: *Solution: Parametric equation of integration path is: Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 17 17

Contour Integral when n=0 when Important Conclusion: integral value is independent to the center point and radius of the circle. Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 18 18

? Contour Integral With Branch Cut 2019/7/8, Page 19 Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 19

Properties of Integral Complex integral has similar properties with definite integral of real variable function. Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 20

Properties of Integral 板书证明 Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 21

Anti-Derivatives 板书证明 Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 22

? Not D but a curve Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 23

Cauchy–Goursat theorem 板书证明 Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 24

Cauchy–Goursat theorem Applications: 板书证明 simple closed contour, closed contours (intersection: finite / infinite) Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 25

Cauchy–Goursat theorem Example: Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 26

Cauchy–Goursat theorem Recall the following theorem Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 27

Cauchy–Goursat theorem 板书证明 Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 28

Cauchy–Goursat theorem principle of deformation of paths Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 29

Cauchy–Goursat theorem Example: ? Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 30

Cauchy Integral Formula 板书证明 Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 31

Cauchy Integral Formula Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 32

Cauchy Integral Formula Gauss's mean value theorem Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 33

Extensions: Analytic Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 34

Extensions: Analytic Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 35

Extensions: Analytic Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 36

Extension: Liouville’s theorem Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 37

Extension: Max Modulus Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 38

Extension: Max Modulus Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 39

Extension: Max Modulus Wei-Shi Zheng wszheng@ieee.org 2019/7/8, Page 40