Page 15 - Calculus with Complex Numbers
P. 15
Ffgure / . 4
'We write
arg (1 + i ) = zr/4 (PV).
For general z = .x + iy we have cos 0 = x(r sin 0 = y/r (see Figure 1.3).
Therefore
= r cos 0 + ir sinp
= rtcos 0 + i sin 0)
io
= r d ,
since, by Taylor's theorem
= cos p + i sin p.
W e call the formula
i0
e = cos p + i sin p
Euler's formula. We call the representation z = reio the polarform for z. We call
the representation z = .x + iy the Cartesian form for z. For example 1 + i
.
2
.$,//--c0,r/4 (see Figure 1.4).
I .7 D e M oivre's theore m
An immediate consequence of Huler's formula (see Section 1.6) is the resultknown
as de M oivre 's theorem viz.
YOS P + i Sin 0)n = (ei P)X = eino = COS l10 + i Sin 110.
