Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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3.3 POLAR REPRESENTATION OF COMPLEX NUMBERS


Rez

Imz

r

θ
x

y z=reiθ

Figure 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+isinnθ

for 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).
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