082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

338 Chapter 9 Complex numbers
andthisrepresentationisvalidforanyrealvalueofx.Theexpressiononther.h.s.is,of
course, an infinite sum but its terms get smaller and smaller, and as more are included,
the sum weobtain approaches e x . Other examples ofpower series include
and
sinx=x− x3
3! + x5
−··· (9.3)
5!
cosx=1− x2
2! + x4
−··· (9.4)
4!
whicharealsovalidforanyrealvalueofx.Itisusefultoextendtherangeofapplicability
of these power series by allowing x to be a complex number. That is, we define the
function e z tobe
e z =1+z+ z2
2! + z3
3! +···
andtheorybeyondthescopeofthisbookcanbeusedtoshowthatthisrepresentationis
valid for all complex numbersz.
We have already seen thatwecan express a complex number inpolar form:
z=r(cosθ+jsinθ)
Using Equations (9.3) and (9.4) wecan write
z =r
{(1 −···)
− θ2
2! + θ4
+j
(θ −···)}
− θ3
4! 3! + θ5
5!
=r
(1 ···)
+jθ− θ2
2! −jθ3 3! + θ4
4! +jθ5 5!
Furthermore, wenote thate jθ can be written as
so that
e jθ =1+jθ+ j2 θ 2
2!
+ j3 θ 3
3!
+···
=1+jθ− θ2
2! −jθ3 3! +···
z=r(cosθ+jsinθ)=re jθ
This is yet another form of the same complex number which we call the exponential
form. We see that
e jθ =cosθ+jsinθ (9.5)
Itis straightforward toshowthat
e −jθ =cosθ−jsinθ