1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

24 CHAPTER l • COMPLEX NUMBERS • EXAMPLE 1.9 Arg (1 + i) = ~· Remark 1.1 Clearly, if z = x + iy = r(cos8 + isin8}, where x =f 0, ...

1.4 • THE GEOMETRY OF COMPLEX NUMBERS, CONTINUED 2 5 In this case, Arg( 4i) = ~ and arg( 4i) = { ~ + 2mr : n is an integer}. As ...

2 6 CHAPTER 1 • COMPl,EX NUMBERS e" = (-1, 0) = - 1 Y~A= (0, l ) = i , ,,-,;i = (! 2' -11) 2 =! 2 + ,qi 2 \ ;-e'°"=ellz=(l,0)= I ...

1.4 • THE GEOMETRY OF COMPLEX NUMBERS, CONTINUED 27 y Figure 1.13 The product of two complex numbers z3 = z1z2. Together with th ...

28 CHAPTER 1 • COMPLEX NUMBERS from the first and second sets, respectively. In this case, arg z 1 + arg z2 = { B1 + B2 : B1 E a ...

4 • THE GEOMETRY OF COMPLEX NUMBERS, CONTINUED 29 •EXAMPLE 1.13 If z = 1 + i , then r = lzl = viz and (} = Argz = ~· Therefore ...

30 C HAPTER 1 • COMPLEX NUMBERS (d) - 2V3 - 2i. (e) c1..'1F · (f) i+G./3. (g) 3 + 4i. (h) (5 + 5i)^3. Show that arg z 1 + arg z ...

1.5 • THE ALOEBRA OF COMPLEX NUMBERS, REVISITED 31 y ~Zl ~) _ Z1 -+------- --x -+-------+-x Figure 1.16 Figure 1.17 1.5 The Alg ...

32 CHAPTER l • COMPLEX NUMBERS EXAMPLE 1.16 Evaluate (-v/3 - i)^30. S olution (- v /;; .> - i)^30 = ( 2e' •(-•"))30 --. = 2 ...

1 .5 • THE ALGEBRA OF COMPLEX NUMBERS, REVISITED 33 Theorem 1.4 implies that if we can find n distinct solutions to the equation ...

34 CHAPTER 1 • COMPLEX NUMBERS Definition 1.12: Primitive nth root For any natural number n, the value Wn given by i a,, 27T.. 2 ...

1.5 • THE ALGEBRA OF COMPLEX NUMBERS, R EVISITED 35 y Figure 1.1 9 The five solutions to t he equation z5 = c. As before, we get ...

36 CHAPTER 1 • COMPLEX NUMBERS y Figure 1.20 The point z =Bi and its three cube roots, zo, zi. and z2. ls the quadratic formula ...

1.5 • THE ALGEBRA OF COMPLEX NUMBERS, REVISITED 3 7 In Exercise 5b of Section 1.2 we asked you to show that a polynomial with no ...

38 CHAP'l'ER l • COMPLEX NUMBERS (b) Use part (a) and De Moivre's formula to derive Lagrange's identity: 1 +cos 8 +cos 2B + · · ...

1.6 • THE TOPOLOGY OF COMPLEX NUMBERS 39 y +--------.x Figure l.Zl The straight-line segment C joining zo to Zt. Sol u t io n Re ...

40 CHAPTER 1 • COMPLEX NUMBERS to 7r, it is on leaf 2; between 7r and^3 ;, it is on leaf 3; and finally, fort between (^3) ; and ...

6 • T HE TOPOLOGY OF COMPLEX NUMBERS 41 y ,,,.. .... -...... I / ' '\ , , I e ~ .\ r---\ ' ·1 J ' ..... __ ,, :; -------x Fi ...

4 2 CHAPTER l • COMPLEX NUMBERS The boundary of DR (zo) is the circle depicted in Figure 1.23. We denote this circle CR. (zo) an ...

1.6 • THE TOPOLOGY OF COMPLEX NUMBERS 43 y -2-~x FigW'e 1.26 The annulus A= {z : I < lzl < 2} is a connected set. in A, as ...