SECTION 2.1 Elementary Number Theory 99
un+2 = −un+1−un, n= 0, 1 , 2 ,...,since the characteristic polynomialC(x) =x^2 +x+ 1 is irreducible
over the real numbers.Assume now that we have the second-order homogeneous linear
difference equation (2.4) has characteristic polynomial with two
complexzerosa+bianda−bi, wherea, b∈R, andb 6 = 0. Using
the same argument in as in the previous section, we may conclude
that acomplexsolution of (2.4) isun=A(a+bi)n, n= 0, 1 , 2 ,...,where Ais any real constant. However, since the coefficients in
the equation (2.4) are real one may conclude that thereal and
imaginaryparts of the above complex solution are also solutions.
Therefore, we would like to find the real and imaginary parts of
the powers (a+bi)n, n ≥ 0. To do this we write the complex
numbera+biintrigonometric form. We start by writinga+bi=√
a^2 +b^2( a
√
a^2 +b^2+
bi
√
a^2 +b^2)
.Next letθbe the angle represented below:!!!!!!!!!√a (^2) +b 2 !!
θ
a
b
Therefore,a+bi= cosθ+isinθ, from which one concludes^25 that
(a+bi)n = (cosθ+isinθ)n= cosnθ+isinnθ.
(^25) This is usually calledDeMoivre’s Theorem, and can be proved by a repeated application of the
addition formulas for sine and cosine.