1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

64 CHAPTER 2 • COMPLEX F UNCTIONS are mapped onto points that lie on the ray p > 0, </> = 2a. If we now restrict the do ...

2.2 • THE MAPPINGS w = z" AND w = zf. 65 y v w=z' x " Figure 2.12 T he mappings w = z^2 a nd z =wt. Solution Using Equations (2- ...

66 CHAPTER 2 • COMPLEX FUNCTIONS y v 2 ib a+ib Figure 2.13 T he transformation w = z^2. What happens to images of regions under ...

2.2 • THE MAPPI NGS w = z" AND w = zk 67 We can use knowledge of the inverse mapping z = w^2 to get further insight into how the ...

68 CHAPTER 2 • COMPLEX FUNCTIONS periodic with period ~, so f is in general an n -to-one function; that is, n points in the z pl ...

2.2 • THE MAPPINGS W =Zn AND W = z!, 69 -------~EXERCISES FOR SECTION 2.2 Find t he images of the mapping w = z^2 in each case, ...

70 CHAP'l'ER 2 • COMPLEX FUNCTIONS Find the image of the sector r > 0, -7' < () <^2 ; under the following mappings. ( ...

2.3 • LIMITS AND CONTINUITY 71 Solution If x = r cos 8 and y = r sin 8 , then 2r^3 cos^3 8 u(x, y) = 2 2 2. 2 = 2rcos^3 8. r COS ...

72 CHAPTER 2 • COMPLEX F UNCTIONS y v .,,..,,..---- - // .... ' ..... w=f(z) Figure 2. 17 The limit f (z)-+ Wo as z-+ zo. J Defi ...

2.3 • LIMITS AND CONTINUITY 73 If we consider w = f (z) as a mapping from the z plane into the w plane and think about the previ ...

7 4 CHAPTER 2 • COMPLEX FUNCTIONS • EXAMPLE 2.17 Show that Jim (z^2 - 2z + 1) = -1. z~l+i S olut ion We let f (z) = z^2 - 2z + 1 ...

2.3 • LIMITS AND CONTI NUITY 75 Definition 2.5 : Continuity of u (x, y) Let u (x , y) be a real-valued function of the two real ...

76 CHAPTER 2 • COMPLEX FUNCTIONS EXAMPLE 2.18 Show that the polynomial function given by is continuous at each point ZQ in the ...

2.3 • L I M ITS AND CONTINUITY 77 One technique for computing limits is to apply Theorem 2.4 to quotients. If we let P and Q be ...

78 CHAPTER 2 • COMPLEX FUNCTIONS (d) z-l+i Jim z2.;!l+- 2 •-2. .;+l ti. (e) z-1+t Jim •',; t •-l-- 2.;+ 2^3 i by factoring. Det ...

2.4 • BRANCHES OF F UNCTIONS 79 Let f (z) = z~ = r~ (cos~+ isin n, where z = re'^8 , r > 0, and -?r < 8 ::5 7r. Use the p ...

80 CHAPTER 2 • COMPLEX FUNCTIONS In our definition of a function in Section 2.1, we specified that each value of the independent ...

2.4 • BRANC HES OF FUNCTIONS 81 so f 1 and h can be thought of as "plus" and "minus" square root functions. T he negative real a ...

82 CHAPTER 2 • COMPLEX FUNCTIONS v w=f,/..z) x z=w' Figure 2. 19 l The branch J .. off (z) = z~. The corresponding branch, denot ...

2.4 • BRANCHES OF FUNCTIONS 83 (a) A portion of D 1 and its image under w = / 1 • (b) A portion of D2 and its image under w = /, ...