Page 19 - Calculus with Complex Numbers

Ffgure / . 8

Note also that I z - w I has a geometric significance as the distance between c
and w on the Argand diagram (see Figure 1.7).

Inequality 3 Iz - v?1 k: IzI- I'r1. This inequality follows from Inequality 1 by

observing that

IzI = I (z - w) + w I :% Iz - 'r1+ I'r1.
Inequality 4
Inequality 5

N.B. Note the minus sign on the right-hand side.

Prove that for all Izl= 2

Solution To prove the right-hand inequality we observe first that

Iz2 + 11s Iz2I+ I11 = 4 + 1 = 5,