While working through some exercises in Walter Ender's Applied Econometric Time Series, I came across an assertion which I struggle to understand.
(for those familiar with the book, I'd like to refer you to "Appendix 1.1: Imaginary Roots and de Moivre’s Theorem").
It is stated, that a homogenous solution to a second order difference equation $y_t=a_1y_{t-1}-a_2y_{t-2}$ can be expressed as $y_t^h = A_1a_1^t + A_2a_2^t$
Characteristic roots are next established using standard procedures.
In this case the discriminant is assumed to be negative. As a result we arrive at the solution using trigonometric identities.
Characteristic roots are found to be $a_1^t=r^t[cos(tθ)+isin(tθ)]$ and $a_2^t=r^t[cos(tθ)-isin(tθ)]$.
Next, it is stated (the part I don't understand) that although $A_1$ and $A_2$ are arbitrary numbers, they must have the form: $A_1 = B_1[ cos(B_2) + i sin(B_2)]$ and $A_2 = B_1[ cos(B_2) – i sin(B_2) ]$ where $B_1$ and $B_2$ are arbitrary real numbers measured in radians.
How come $A_1$ and $A_2$ "must have" this form? Can't they simply remain $A_1$ and $A_2$, as it is in the case when the discriminant is positive?
Yes they can, but they are now complex but not independent. Indeed, complex numbers carry two constants (the real and the imaginary part), for a total of four, while the solution of a second order difference equation only depends on two.
For the solution to be real, it must equal its complex conjugate, so that
$$A_1r^t[\cos(tθ)+i\sin(tθ)]+A_2r^t[\cos(tθ)-i\sin(tθ)]=\\(A_1r^t[\cos(tθ)+i\sin(tθ)]+A_2r^t[\cos(tθ)-i\sin(tθ)])^*= \\A_1^*r^t[\cos(tθ)-i\sin(tθ)]+A_2^*r^t[\cos(tθ)+i\sin(tθ)].$$
By identification, $A_1=A_2^*,A_2=A_1^*$ hence the two complex constants are conjugate of each other. The $B$ representation is just their polar form.