1550251515-Classical_Complex_Analysis__Gonzalez_

58

Thus the property lzwl = lzlw holds only when w is real.

EXERCISES 1.6

1. Compute the following.
   (a) (--1)1/4 (b) i-2/3
   (d) (--i)3/2 (e) e-i+i11-/6

(g) el-i (h) e4+i11"f2

(c) (1 + i)^112

(f) e2+3i

(i) log i

Chapter 1

(j) log( -4) (k) log(l. - i)

(m) Log( v'3 -i) (n) 10Hi

(1) Log(-3 + 3i)

(o) (-i)Hi

(p) (:t+i)-^2 i (q) (p.v.)(1+i); (r) (p.v.) ii

2. Show that:
   (a) ez = ez


3. Find all values of z such that:
   (a)ez=2 (b)ez=l-iVs


4. Show that ez^1 = ez^2 iff z 1 - z 2 = 2bri (k an integer).


5. If z =f 0, show that I (p.v.) zil < e,,..


6. Prove that the set {zw^1 zw^2 } contains a set {zw^1 +w^2 }


7. Let z1 and Z2 be complex numbers such that 0 < lz1 I
   lz2 I ::; 1. Prove that

lz1 - z2 I ::; I log z1 -log z2 I

::; 1 and 0 <

[T. Hern, Amer. Math. Monthly 79 (1972), 1035-1037]

1.17 INFINITY IN THE COMPLEX PLANE. THE

RIEMANN SPHERE

For later developments it is convenient to adjoint to the complex num-

ber system C one ideal element or improper complex number (an ideal or

improper point in the geometric language), denoted by the symbol oo.

Definition 1.11 By the extended complex number system C* we mean

the union of C with a symbol oo that satisfies the following properties:

1. For z E C,

z + 00 = 00 + z = z - 00 = oo, _!_ =^0

00

2. For z E C, z =f O,

z(oo) = (oo)z = oo,

z

• = 00
  0

3. (oo)(oo) = oo