Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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8 CHAPTER 1. THE COMPLEX NUMBERS<br />
and the trigonometric angle sum formulae. Notice that<br />
(cosθ,sinθ)·(cosφ,sinφ) = (cosθcosφ−sinθsinφ,cosθsinφ+sinθcosφ)<br />
= (cos(θ+φ),sin(θ+φ)).<br />
Thatis, multiplication<strong>of</strong>2complexnumbersontheunitcirclex 2 +y 2 = 1corresponds<br />
to addition <strong>of</strong> their angles <strong>of</strong> inclination to the x axis. In particular, the mapping<br />
f(z) = z 2 doubles the angle <strong>of</strong> z = (x,y) and f(z) = z n multiplies the angle <strong>of</strong> z<br />
by n. These statements hold even if z lies on a circle <strong>of</strong> radius r �= 1:<br />
this is known as deMoivre’s Theorem.<br />
(rcosθ,rsinθ) n = r n (cosnθ,sinnθ);