1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 45 are said to be square roots, for n = 3 the roots are said to be cubic roots, for n = 4 fourth roots, and so o ...

46 From (l..12-2) and (1.12-5) we obtain x^2 = %(r +a), ;; = ± ./%(r +a), Chapter 1 y^2 = %(r - a) y = ± .;---1/2-( r---a-) Thes ...

Complex Numbers 47 where, as before, k denotes an arbitrary integer. This shows that the modulus of any nth root of w "is unique ...

48 Chapter^1 w y x Fig. 1.13 it is clear that the nth roots of w can be obtained by multiplying the principal. root of w by each ...

Complex Numbers (e) (-2 + 3i)^3 (g) [ 3 (cos i + i sin i ) r (i) (2ei1r/4)3 Simplify. (f) (-1+2i)^4 (h) [ ~ (cos i + i sin i) ...

50 Chapter 1 for z #- 1 and derive the formulas ~ k() _ ~ sin(n + 1/i)() 6 cos - 2 + 2. 1; () k=O Sln 2 ~. k() _ ~ ~ () _ cos(n ...

Complex Numbers 51 20. (a) If n is a positive integer, show that cos nB = Pn( cos B) where Pn(x) is a polynomial of the nth degr ...

52 Chapter^1 since the right-hand side means { rmfnei(m1J+2krr)/n : k = O, 1, ... , n _ 1} which coincides with (1.13-2) iff m a ...

Complex Numbers 53 while 2(*V-1) = {2i, -2i} However, for principal values it is true that vz+ vz=2vz the symbol+ denoting now o ...

54 Chapter^1 Proofs Property 1 follows at once from (1.14-1). To prove property 2 we have, using (1.14-1), the additivity of rea ...

Complex Numbers 55 where ln r indicates, as usual, the natural or N aperian logarithm of the positive real number r. Thus the se ...

56 from which we derive, by Theorem 1.11, elog z+log z' = zz', elog z-log z' = z z' Chapter 1 This shows that a value of log( zz ...

Complex Numbers 57 zn has just one value, which coincides with the one defined in Section 1.11; and when w is an irreducible fra ...

58 Thus the property lzwl = lzlw holds only when w is real. EXERCISES 1.6 Compute the following. (a) (--1)1/4 (b) i-2/3 (d) (-- ...

Complex Numbers 59 Expressions such as oo + oo, oo - oo, 0 · oo, or oo/oo are not defined since they do not lead to meaningful r ...

60 Chapter^1 To determine analytically the rectangular coordinates (a, /3, 'Y) of the point Q corresponding under stereographic ...

Complex Numbers 61 Equations (1.17-9) together with (1.17-8) determine a, (3, 'Yin terms of z. Conversely, equations (1.17-3) or ...

62 Chapter 1 If we compute the distances NP and NQ from N(O,O, 1) to the points P(x,y,0) and Q(a,/3,"(), respectively, we find t ...

Complex Numbers 63 The length Q' Q of the chord joining Q and Q' is the Euclidean distance between the spherical images of z and ...

64 Chapter^1 great circle of the Riemann sphere that contains those two points. This arc is well determined except when the poin ...