1550251515-Classical_Complex_Analysis__Gonzalez_

Topology of Plane Sets of Points 89

EXERCISES 2.2

Describe graphically the sets of points in the complex plane defined by

the following equalities or inequalities.

1. Re z ~ 0 2. Im z < 0

3. lzl = 2 4. lz - 1 I < 1

5. z = 3 + (1 + i)t, t E ~ 6. z = (1 - t)i + t, 0 $ t $ 1


6. (z - i)(z + i) = 4 8. z + 1 = 3eit, 0 St S 271"
   7r (z -2)


7. Argz = "4 10. Im --i-> 0
   -71" 7r k


8. 2 < Argz < "4 12. 0 < Argz < "4


9. {z: lzl > 1, IArgzl Si} 14. Re(~)= 2

15. lz + ll = lz - 21 16. a< Rez < b, a, b real

17. 2Slz-ilS3 18. lz-llSlz+ll

19. lz -11S2lz + ll 20. lzl - Rez S^1 / 2

21. Im (z ~ l) = 0 22. lz^2 - ll = 1

23. Arg( z - z 0 ) = a, a const. 24. Im z^2 = 1

25. lz - 1 I + lz + 1 I = 4 26. Im ( ~) > ~

27. Apply condition (2.4-10) to show that the straight line defined by the

equation Az + Az + C = 0, O, A #-0, C real, is perpendicular to the

vector A. Hint: Note that if z 1 and z 2 (z1 #-z2) are two points on the

lil\e, the vector z 2 - z 1 is in the direction of the line.

28. Show that the equation of the straight line through the points z 1 and
    z2 (z1 #-z2) can be written as

z=z 1 +t(z 2 -z 1 ), tE~

or alternatively, as

29. If z 0 #-0, show that Re(z/zo) = 1 represents a straight line, z 0 being

the foot of the perpendicular from the origin to the line.

30. Prove that the equation of the straight line through the points z 1 and
    Zz (z1 #-z2) can be put in the form

1 1 1

Z Z1 Zz = 0.

z z1 .Z2