3.3 POLAR REPRESENTATION OF COMPLEX NUMBERS
RezImzrθ
xy z=reiθFigure 3.7 The polar representation of a complex number.From (3.19), it immediately follows that forz=iθ,θreal,eiθ=1+iθ−θ^2
2!−iθ^3
3!+··· (3.21)=1−θ^2
2!+θ^4
4!−···+i(
θ−θ^3
3!+θ^5
5!−···)
(3.22)and hence that
eiθ=cosθ+isinθ, (3.23)where the last equality follows from the series expansions of the sine and cosine
functions (see subsection 4.6.3). This last relationship is calledEuler’s equation.It
also follows from (3.23) that
einθ=cosnθ+isinnθfor alln. From Euler’s equation (3.23) and figure 3.7 we deduce that
reiθ=r(cosθ+isinθ)=x+iy.Thus a complex number may be represented in the polar form
z=reiθ. (3.24)Referring again to figure 3.7, we can identifyrwith|z|andθwith argz.The
simplicity of the representation of the modulus and argument is one of the main
reasons for using the polar representation. The angleθlies conventionally in the
range−π<θ≤π, but, since rotation byθisthesameasrotationby2nπ+θ,
wherenis any integer,
reiθ≡rei(θ+2nπ).