Chapter 3. Elementary Functions Weiqi Luo ( 骆伟祺 ) School of Software Sun Yat-Sen University ： Office ： # A313

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

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

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

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.

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

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

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

School of Software pp Ex. 1, Ex. 6, Ex Homework 9

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

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

School of Software  Example The Logarithmic Function 12

School of Software  Example 2 & The Logarithmic Function 13

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

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

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

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

School of Software pp Ex. 1, Ex. 3, Ex. 4, Ex. 9, Ex Homework 18

School of Software 32. Some Identities Involving Logarithms 19 where

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

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

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

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

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.

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

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

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.

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

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.

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

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

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

School of Software pp Ex. 2, Ex Homework 33

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

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

School of Software pp Ex Homework 36

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.

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

School of Software pp Ex Homework 39