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a. What geometric shape does the polygon with the solutions as ver-
tices form?
b. What is the sum of these roots? (Derive your answer both algebra-
ically and geometrically.)

6.4.3 The Function y = e jθθ θθ
As alluded to previously, the expression cos(θ) + j sin(θ) behaves very much
as if it was an exponential; because of the additivity of the arguments of each
term in the argument of the product, we denote this quantity by:
j θ
e = cos(θ) + j sin(θ) (6.35)
PROOF Compute the Taylor expansion for both sides of the above equation.
jθ
The series expansion for e is obtained by evaluating Taylor’s formula at x =
jθ, giving (see appendix):

∞
jθ
e jθ = ∑ 1 () n (6.36)
n=0 n!
j θ
When this series expansion for e is written in terms of its even part and odd
part, we have the result:

∞
∞
jθ
jθ
e jθ = ∑ 1 () 2 m + ∑ 1 () 2 m+1 (6.37)
m=0 2 ( m)! m=0 2 ( m + 1)!
2
However, since j = –1, this last equation can also be written as:
∞
∞
e jθ = ∑ − ( 1 ) m θ () 2 m + j ∑ − ( 1 ) m θ () 2 m+1 (6.38)
1
m=0 (2 m)! m=0 (2 m + )!
which, by inspection, can be verified to be the sum of the Taylor expansions
for the cosine and sine functions.
j(θ 1 θ+
In this notation, the product of two complex numbers z and is: r r e 2 ) .
z
1 2 1 2
It is then a simple matter to show that:
r
If: z = exp( )θ (6.39)
j
r
Then: z = exp( j − )θ (6.40)

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