30 C HAPTER 1 • COMPLEX NUMBERS
(d) - 2V3 - 2i.(e) c1..'1F ·
(f) i+G./3.
(g) 3 + 4i.
(h) (5 + 5i)^3.- Show that arg z 1 + arg z2 ~ arg z 1 z2, thus completing the proof of Theorem 1.3.
- Express the following in a + ib form.
(a) elf.
(b) 4e-i~.(c) Se i'• T.
(d) -2e•~.
( e) 2ie-•'lf.
· h.
(f) 6e' • e'".
(g) e^2 e^1 ".
(h) e'7 e-i•.
Show t hat arg z 1 = arg z 2 iff z 2 = cz 1 , where c is a positive real constant.
Let z 1 = - 1 + iVS and z 2 = -VJ + i. Show that the equation
Arg(z1Z2) = Argz1 + ArgZ2 does not hold for t he specific choice of z 1 and z2.
Show that the equation Arg(z 1 z2) = Arg z1 + Arg z2 is true if
- 2 • < Argz1 ::; I and - 2 " < Argz2 ::; ~· Describe the set of points that meets
this criterion.
- 2 • < Argz1 ::; I and - 2 " < Argz2 ::; ~· Describe the set of points that meets
Describe the set of complex numbers for which Arg(±) # - Arg{z). Prove your
assertion.
Establish the identity arg( ~) = arg z1 - arg z2.
Show t hat arg( ~) = -arg z.
Show t hat arg(z1z2) = arg z 1 - arg z2.
Show that if z # 0, then
(a) Arg(zz) = 0.
(b) Arg(z + z) = O when Re(z) > O.- Let z1, z2, and z3 form the vertices of a triangle as indicated in Figure 1.16. Show
t hat°' E arg (~~=~:) = arg(z2 -z1) - arg (z3 - z1) is an expression for the angle
at the vertex z 1.1 5. Let z # zo. Show that the polar representation z -z 0 = p( cos</> + i sin qi) can be
used to denote t he displa.cement vector from zo to z, as indicated in Figure 1.17.