✔ 最佳答案
(1a) if z + 1 / z = -8 + 9i , simplify z^2 + 1 / z^2
Square both sides:
(z + 1/z)^2 = z^2 + 1/z^2 + 2
= (-8 + 9i)^2
= 64 - 81 - 144i
= -17 - 144i
=> z^2 + 1/z^2 = -19 - 144i
(2b) z^3 + 1 / (z^3) = ?
= (z + 1/z)*(z^2 - 1 + 1/z^2)
= (-8 + 9i)*(-19 - 144i - 1)
= (8 - 9i)*(20 + 144i)
= 4*(8 - 9i)*(5 + 36i)
= 4*(40 + 9*36 - 45i + 8*36i)
= 4*(364 + 243i)
(2) Let a and z be a real number and z = [(a-i) / (1+i)]^2 . a = ?
z = (a-i)^2/(1+i)^2
= (a^2 - 2ai - 1)/(1 - 2i - 1)
= [(a^2-1)-2ai]/(-2i)
= [(a^2-1)-2ai]i/(-2i^2)
= [2a + (a^2-1)i]/2
= a + (a^2-1)i/2
= real
=> a^2 = 1
=> a = +-1
=> a = z = +-1
