1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

44 CHAPTER 1 • COMPLEX NUMBERS EXAMPLE 1.26 Show that the right half-plane H = {z: Re (z) > O} is a domain. Solution First ...

1.6 • THE TOPOLOGY OF COMPLEX NUMBERS 45 Figure 1.28 Are z 1 and z2 in the interior or exterior of this simple closed curve? The ...

46 CHAPTER 1 • COMPLEX NUMBERS Sketch the curve z (t) = t^2 + 2t + i (t + 1) (a) for -1 $. t $. 0. (b) for 1 $. t $. 2. Hint: ...

1.6 • THE TOPOLOGY OF COMPLEX NUMBERS 47 Let S = { z1, z2, ... , Zn} be a finite set of points. Show that S is a bounded set. P ...

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Overview The last chapter developed a basic theory of complex numbers. For the next few chapters we turn our attention to functi ...

50 CHAPTER 2 • COMPLEX FUNCTIONS w =fl.z) = u +iv u = u(x, y) v=v(x.y) Figure 2.1 The mapping w = f (z). v Just as z can be expr ...

2.1 • FUNCTIONS AND L INEAR MAPPINGS 51 Examples 2.1 and 2.2 show how to find u(x, y) and v(x, y) when a rule for computing f is ...

52 CHAPTER 2 • COMPLEX FUNCTIONS Once we have defined u and v for a function f in Cartesian form, we must use different symbols ...

2.1 • FUNCTIONS AND LINEAR MAPPINGS 53 y v w = f(Z) = 11 + iv Figure 2.2 f maps A onto B; f maps A into R. You will learn in Cha ...

54 CHAPTER 2 • COMPLEX FUNCTIONS y !<z> = z^2 v ........ ........................ -........ -.. .. .... .... .. .. ,..... ...

2.1 • FUNCTIONS AND LINEAR MAPPINC S 55 We now show how to find the image B of a specified set A under a given mapping u+iv = w ...

56 CHAPTER 2 • COMPLEX FUNCTI ONS y W=t+B u=x+a v=y+b ~-·· -·/B=~v+=i~t) I I I I '---- F igure 2.5 The t ranslation w = T (z) ...

2.1 • FUNCTIONS AND LINEAR MAPPINGS 57 1 (a) 2 1.5 I 0.5 0 ~::::'.::~.......+-- 0.5 I 1.5 2 2.5 3 3.5 (b) Figure 2.7 (a) Plot ...

58 CHAPTER 2 • COMPLEX FUNCT IONS y Ki w=Kz u=Kx v i .. v=Ky i -+~~+-~~~-+--X U K K Figure 2.8 The magnificationw= S(z) = K z = ...

2.1 • FUNCTIONS AND LINEAR MAPPINGS 59 (Method 2): When we write w = f (z) in Cartesian form as w = u + iv = i(x +iy) + i = -y+i ...

60 CHAPTER, 2 • COMPLEX FUNCTIONS Solution The inverse transformation is z = w3~4;2', so if we designate the range of f as B, th ...

2.1 • FUNCTIONS AND LINEAR. MAPPINGS 61 y "'=f(l) -- Figure 2.11 The mapping w = f (z) = (-1 + i) z- 2 + 3i. -------~EXERCISES F ...

62 CHAPTER 2 • COMPLEX FUNCTIONS (c) f (i^2 ;). (d) /(2+i7r). (e) f (37ri). (f) Is f a one-to-one function? Why or why not? For ...

2.2 • THE MAPPINGS w =Zn AND w = zfi 63 (b) Show, a.dditionally, that if g is on~to-one from B onto S, then h ( z) is one-to-one ...