Figure 10-9. Polar form of complex numbers.
In this figure ris called the modulus or length of the complex number λand θis
called an argument of the complex number λ. Values of 2 πcan be added to obtain
other arguments of λ. A value of θsatisfying −π < θ ≤πis called the principal value
of the argument of λ. The complex root λ 1 has a modulus and argument of
r=
√
52 + 7^2 and θ= arctan(7 /5). (10 .59)
The polar form of the characteristic roots produce solutions to the difference equa-
tions that can then be expressed in the form
y 1 (n) = rneinθ and y 2 (n) = rne−inθ.
The Euler’s identity eiθ = cos θ+isin θ, is used to write these solutions in the form
y 1 (n) = rn(cos nθ +isin nθ )
y 2 (n) = rn(cos nθ +isin nθ ).
The solutions y 1 (n)and y 2 (n)are independent solutions of the given difference equa-
tion and hence any linear combination of these solutions is also a solution. Form
the linear combinations
y 3 (n) =^12 [y 1 (n) + y 2 (n)] and
y 4 (n) =^1
2 i
[y 1 (n)−y 2 (n)]
