1550251515-Classical_Complex_Analysis__Gonzalez_

34

so that

and

Note If 81 = Arg z1, 82 = Arg z2, then

Z1

Arg - = Argz1-Argz2+27rn(z1,z2)

Z2

where

Exercises 1.3

if - 7l" < Arg z 1 - Arg z2 :::; 7l"

if - 271" < Argz1 -Argz2:::; -7!"

if 7l" < Argz1 -Argz2 < 27!"

1. Represent geometrically the following complex numbers.
   (a) 3i (b) -7
   (c)3+i (d)l-i
   (e) -J3 + i (f) % + %J3i

Chapter 1

2. Write the following complex numbers in polar form and also m
   exponential form.
   (a)l+i (b)-5
   (c) -i (d) -^1 / 2 - %J3i
   (e) -3 + 3i (f) (1 + -15) + V~~o---2-V5-5i


3. Write the following numbers in binomial form.

(a) 3( cos 7l" + i sin 7l") (b) 4 (cos

3

; + i sin

3

; )

(c) v2(cos i +isin i) (d) 2ei7T:/^2

,(e) 5J2ei37T:/4 (f) 4e-i57T:/6

4. Perform the indicated operations.

()5( a cos 3 7l" + i •• sm 71") 3 x 2 ( cos '2 7l" + i •• sm '2 71")

(b) (^4) ( cos '6 7l" + sm • 7l" '6 ) x 3 ( cos 571".. 571" )
6

• ism 6
  ()4( c cos "3 7l" + i •• sm "3 71") + 2 ( cos 7l" •• 71")
  4


• i sm
  4
  ( d) 20 (cos
  5
  6
  7!" + i sin
  5

; ) + 8( cos 7l" + i sin 7l")

5. Use the exponential form to perform the following operations.

(a) (V3-i)(l+i) (b) (1-iV3)^3