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 Wei-Shi Zheng wszheng@ieee.org 16.09.2018, Page 2

Exponential Function Definition: 3

Exponential Function Equivalent between two complex variables Wei-Shi Zheng wszheng@ieee.org 16.09.2018, Page 4

Exponential Function Roots root 16.09.2018, Page 5 Wei-Shi Zheng wszheng@ieee.org 16.09.2018, Page 5

Exponential Function Sum Rule: The definition of exponential function is equivalent to the expression: Sum Rule: 6

Exponential Function >0

Exponential Function *Example 1: *Solution: 8

Exponential Function 9

Exponential Function Example 2: Compute the principal angles Solution: Give the principal value of argument: Solution: 10

Exponential Function 11

Exponential Function Example 3: Solution: 12

Exponential Function Example 4: ?

Logarithmic Function 1.Definition: 14

Logarithmic Function

Logarithmic Function

Logarithmic Function

Logarithmic Function Review

Logarithmic Function

Logarithmic Function Branch

Logarithmic Function Branch Cut

Logarithmic Function branch cut(including the origin) f(z) singular points F(z) branch

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 real variable logarithmic function. 23

Logarithmic Function *Example 5: *Solution: 24

Logarithmic Function *Example 6: *Solution: 25

Logarithmic Function 26

Logarithmic Function Properties: 27

Logarithmic Function Properties: 4) 5) 6) 28

Complex Exponents any complex number Definition

Complex Exponents

Complex Exponents Example