Complex Numbersby repeated application of Theorem 1. 7, we find thatZ1Z2 ... Zn= (r1r2 ... rn)ei(81+82+··+8n)
43In particular, for ri = rz = · · · = rn = r and 81 = 82 = · · · = 8n = 8,
we have(1.11-2)
or
Zn = rn( COS n8 + i sin n8) (1.11-3)If z = ei^8 , we get
or
(cos e + i sin er = cos ne + i sin ne (1.11-4)
This is known as De Moivre's theorem (A. De Moivre, 1707).Equation (1.11-2) is also valid for n = 0 and n = 1. It also holds for n
a negative integer. In fact, if n = -m, m being a positive integer, we have
n -m 1 1 = r-me-im8 = rnein8
z = z = zm = rmeim8Examples
- ( 2 ei1r/3)4 = 16 e4i1r/3
2. (4ei'lr/^5 )-^3 = (1/64)e-^3 i'lr/^5
Powers with complex exponents (in particular, with real exponents) are
discussed in Section 1.16.
Note From (1.11-3) it may seem that
argzn = ne = n argzHowever, the right side does not give in general all the values of arg zn.
For instance, if n = 3, z = i, we have
argi^3 = arg(-i) = -% + 2k17r
and3 arg i = 3 ( % + 2k27r) =
3
; + 6k 2 7rwhere k 1 and k 2 are arbitrary integers. For ki = 0 we get
Argi^3 = -~
2