1. Chapter 3. Elementary Functions Weiqi Luo ( 骆伟祺 ) School of Software Sun Yat-Sen University Email ： weiqi.luo@yahoo.com Office ： # A313 weiqi.luo@yahoo.com

2. School of Software  The Exponential Functions  The Logarithmic Function  Branches and Derivatives of Logarithms  Some Identities Involving Logarithms  Complex Exponents  Trigonometric Function  Hyperbolic Functions  Inverse Trigonometric and Hyperbolic Functions 2 Chapter 3: Elementary Functions

3. School of Software  The Exponential Function 29. The Exponential Function 3 According to the Euler’ Formula Note that here when x=1/n (n=2,3…) & y=0, e 1/n denotes the positive nth root of e. u(x,y) v(x,y) Single-Valued

4. School of Software  Properties 29. The Exponential Function 4 Let Real value: Refer to pp. 18

5. School of Software  Properties 29. The Exponential Function 5 Refer to Example 1 in Sec 22, (pp.68), we have that everywhere in the z plane which means that the function e z is entire.

6. School of Software  Properties 29. The Exponential Function 6 For any complex number z which means that the function e z is periodic, with a pure imaginary period of 2πi

7. School of Software  Properties 29. The Exponential Function 7 For any real value x while e z can be a negative value, for instance

8. School of Software  Example In order to find numbers z=x+iy such that 29. The Exponential Function 8

9. School of Software pp. 92-93 Ex. 1, Ex. 6, Ex. 8 29. Homework 9

10. School of Software  The Logarithmic Function 30. The Logarithmic Function 10 It is easy to verify that Please note that the Logarithmic Function is the multiple-valued function. … One to infinite values

11. School of Software  The Logarithmic Function 30. The Logarithmic Function 11 Suppose that  is the principal value of argz, i.e. -π <  ≤π is single valued. And

12. School of Software  Example 1 30. The Logarithmic Function 12

13. School of Software  Example 2 & 3 30. The Logarithmic Function 13

14. School of Software  The Logarithm Function where=Argz, is multiple-valued. If we let θ is any one of the value in arg(z), and let α denote any real number and restrict the value of θ so that The above function becomes single-valued. With components 31. Branches and Derivatives of Logarithms 14

15. School of Software  The Logarithm Function is not only continuous but also analytic throughout the domain 31. Branches and Derivatives of Logarithms 15 A connected open set

16. School of Software  The derivative of Logarithms 31. Branches and Derivatives of Logarithms 16

17. School of Software  Examples When the principal branch is considered, then 31. Branches and Derivatives of Logarithms 17 And

18. School of Software pp. 97-98 Ex. 1, Ex. 3, Ex. 4, Ex. 9, Ex. 10 31. Homework 18

19. School of Software 32. Some Identities Involving Logarithms 19 where

20. School of Software  Example 32. Some Identities Involving Logarithms 20

21. School of Software 32. Some Identities Involving Logarithms 21 When z≠0, then Where c is any complex number

22. School of Software pp. 100 Ex. 1, Ex. 2, Ex. 3 32. Homework 22

23. School of Software  Complex Exponents When z≠0 and the exponent c is any complex number, the function z c is defined by means of the equation where logz denotes the multiple-valued logarithmic function. Thus, z c is also multiple-valued. 33. Complex Exponents 23 The principal value of z c is defined by

24. School of Software 33. Complex Exponents 24 If and α is any real number, the branch Of the logarithmic function is single-valued and analytic in the indicated domain. When the branch is used, it follows that the function is single-valued and analytic in the same domain.

25. School of Software  Example 1 33. Complex Exponents 25 Note that i -2i are all real numbers

26. School of Software  Example 2 The principal value of (-i) i is 33. Complex Exponents 26 P.V.

27. School of Software  Example 3 The principal branch of z 2/3 can be written 33. Complex Exponents 27 Thus This function is analytic in the domain r>0, -π<  < π P.V.

28. School of Software  Example 4 Consider the nonzero complex numbers 33. Complex Exponents 28 When principal values are considered

29. School of Software  The exponential function with base c 33. Complex Exponents 29 When logc is specified, c z is an entire function of z. Based on the definition, the function c z is multiple-valued. And the usual interpretation of e z (single-valued) occurs when the principal value of the logarithm is taken. The principal value of loge is unity.

30. School of Software pp. 104 Ex. 2, Ex. 4, Ex. 8 33. Homework 30

31. School of Software  Trigonometric Functions 34. Trigonometric Functions 31 Here x and y are real numbers Based on the Euler’s Formula Here z is a complex number

32. School of Software  Trigonometric Functions 34. Trigonometric Functions 32 Both sinz and cosz are entire since they are linear combinations of the entire Function e iz and e -iz

33. School of Software pp.108-109 Ex. 2, Ex. 3 34. Homework 33

34. School of Software  Hyperbolic Function 35. Hyperbolic Functions 34 Both sinhz and coshz are entire since they are linear combinations of the entire Function e iz and e -iz

35. School of Software  Hyperbolic v.s. Trgonometric 35. Hyperbolic Functions 35

36. School of Software pp. 111-112 Ex. 3 35. Homework 36

37. School of Software 36. Inverse Trigonometric and Hyperbolic Functions 37 In order to define the inverse sin function sin -1 z, we write When Multiple-valued functions. One to infinite many values Similar, we get Note that when specific branches of the square root and logarithmic functions are used, all three Inverse functions become single-valued and analytic.

38. School of Software  Inverse Hyperbolic Functions 36. Inverse Trigonometric and Hyperbolic Functions 38

39. School of Software pp. 114-115 Ex. 1 36. Homework 39