Engineering Mathematics Complex Variables Applications Chapter 3 wszhengieee

Engineering Mathematics Complex Variables & Applications Chapter 3 郑伟诗 wszheng@ieee. org, http: //sist. sysu. edu. cn/~zhwshi/

Outline • Exponential Function • Logarithmic Function • Complex Exponents

Exponential Function Definition: Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 3 3

Exponential Function • Equivalent between two complex variables

Exponential Function • Roots root

Exponential Function The definition of exponential function is equivalent to the expression: Sum Rule: Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 6 6

Exponential Function Wei-Shi Zheng wszheng@ieee. org >0 11/25/2020, Page 7

Exponential Function *Example 1: *Solution: Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 8

Exponential Function Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 9 9

Exponential Function Example 2: Compute the principal angles Give the principal value of argument: Solution: Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 10 10

Exponential Function Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 11 11

Exponential Function Example 3: Solution: Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 12

Exponential Function Example 4: ? Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 13

Logarithmic Function 1. Definition: Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 14 14

Logarithmic Function Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 15

Logarithmic Function Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 16

Logarithmic Function Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 17

Logarithmic Function Review Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 18

Logarithmic Function Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 19

Logarithmic Function • Branch Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 20

Logarithmic Function • Branch Cut Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 21

Logarithmic Function branch cut(including the origin) f(z) singular points F(z) branch Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 22

Logarithmic Function *Example 4: *Solution: *Remark: Among Functions of Real Variable, negative numbers do not have logarithm, yet complex variable logarithmic function is the extension of Wei-Shi Zheng 11/25/2020, wszheng@ieee. org real variable logarithmic function. Page 23 23

Logarithmic Function *Example 5: *Solution: Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 24 24

Logarithmic Function *Example 6: *Solution: Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 25 25

Logarithmic Function Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 26 26

Logarithmic Function Properties: Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 27 27

Logarithmic Function Properties: 4) 5) 6) Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 28 28

Complex Exponents any complex number Definition Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 29

Complex Exponents Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 30

Complex Exponents Example Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 31

Wei-Shi Zheng wszheng@ieee. org 11/25/2020, Page 32