Page 18 - Calculus with Complex Numbers

Ffgure /.5

I . I 0 rlth roots of u nity

Just like any other non-zero complex number 1 has n complex rlth roots. W e have
1 = c 2=i = c
4xf= . . .
0 = c
1 0 zxi/n 4xf/n
1/n = c
, c , c , . . .
lf we denote û? = elKi/n then the n rlth roots of 1 are 1 , û?, û) 2 . . . û?Fl-1
(see Figtlre 1.5 for the case n = 8). We call û? tjmprimitive rlth root of 1.
N.B. 11/n = 1 (PV) of cotlrse.

Lemm a 1 + û? + *2 + . . . + û?n-1 = O

I . I I I n e q u al i ti e s

The fundamental inequality is the so called triangle or parallelogram inequality
and is as follows.

Inequality 1 I z + w I1qL Iz I + Iw I. This inequality expresses the fact that the
diagonal of a parallelogram has length less than or equal to the sum of the lengths
of two adjacent sides (see Figure 1.6). Hquivalently, that the length of one side of
a triangle is less th= or equal to the sum of the lengths of the other two sides.
(Consider the triangle with vertices 0, z, z + w.)

Inequality l Iz - M?11qL IzI + IJ?1 . This inequality follows from Inequality 1 by
putting -w for w .
N.B. Note the plus sign on the right-hand side.