1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

4 CHAPTER 1 • COMPLEX NUMBERS Since (2 + J=l)^3 = 2 + nJ=I, we clearly have 2 + A = {12 + UJ=l. Similarly, Bombelli showed that ...

1.1 • THE ORIGIN OF COMPLEX NUMBERS 5 b^2 - c^2 < 0. He evidently thought that, because b is shorter than c, it could no long ...

6 CHAPTER l • COMPLEX NUMBERS firm logical foundation was crucial, but so, too, was a willingness to modify some ideas concernin ...

1.2 • THE ALGEBRA OF COMPLEX NUMBERS 7 1. 2 The Algebra of Complex Numbers vVe have shown that complex numbers came to be viewed ...

8 CHAPTER 1 • COMPLEX N UMBERS EXAMPLE 1.1 If z 1 = (3, 7) and Z2 = (5, -6), then z1 + Z2 = (3, 7) + (5, -6) = (8, 1) and ZJ - ...

2 • THE ALGEBRA OF COMPLEX NUMBERS 9 We get the same answer by using the notation z 1 = 3 + 7i and z2 = 5 - 6i: Z1Z2 = (3, 7)( ...

10 CHAPTER l • COMPLEX NUMBERS • EXAMPLE 1.3 If z1 = (3, 7) and z2 = (5, -6), then ZJ = (3,7) = (15- 42 18 +35) = ( - 27 53)· z2 ...

1.2 • THE ALGEBRA OF COMPLEX NUMBERS 11 (P3) Addit ive identity: There is a complex number w such that z +w = z for all complex ...

1 2 CHAPTER 1 • COMPLEX N UMBERS X1X2 (x1, O)(x2, 0) (by our agreed correspondence) (X 1 X2 - 0, 0 + 0) (by definit ion of multi ...

1.2 • THE ALCEBR.A OF COMPLEX NUMBERS 13 Definition 1.5: Real part The real part of z, denoted Re (z), is the real number x. ~~~ ...

14 CHAPTER 1 • COMPLEX NUMBERS Because of what it erroneously connotes, it is a shame that the term imagi- nary is used in Defin ...

1.2 • THE ALGEBRA OF COMPLEX NUMBERS 15 (i) Re [(x + iy) (x -iy)J. {j) Im [(x + iy)^3 ]. Show tha.t zz is always a. real number ...

16 CHAPTER 1 • COMPLEX NUMBERS Verify that if z = (x, y), with x and y not both 0, then z-^1 = <\^0 > (i.e., z -^1 = ~}. ...

1.3 • THE GEOMETRY OF COMPLEX NUMBERS 17 y 4 +3i 4 Figure 1.5 The difference z1 - z2. Figure 1.6 The real and imaginary parts of ...

1 8 CHAPTER 1 • COMPLEX NUMBERS y (0, y) P = (x, y) = z llm(z}I !Re(z)I +----~--1--x 0 = (0, 0) Q = (x, 0) Figure 1- 7 The modul ...

1.3 • THE GEOMETRY OF COMPLEX NUMBERS 19 EXAMPLE 1.5 To produce an example of which Figure 1.9 is a reason- able illustration, ...

20 CHAPTER l • COMPLEX N UMBERS y Figure 1.10 The geometry of mul t iplicat ion. -------~EXERCIS ES FOR SECTION 1. 3 Evaluate t ...

1.3 • THE GEOMETRY OF COMPLEX NUMBERS 21 Sketch the sets of points determined by the following relations. (a) lz + 1 - 2il = 2 ...

22 CHAPTER l • COMPLEX NUMBERS Let z 1 and z 2 be two distinct points in the complex plane, and let K be a positive real consta ...

1.4 • THE GEOMETRY OF COMPLEX NUMBERS, CONTINUED 23 • EXAMPLE 1.7 If z = 1 + i, then r = J2 and z = ( J2cos f, J2sin f) = J2(cos ...