It is not the intention here to present a new method for finding the roots of a cubic polynomial; rather, the aim is to allow all the roots to be visualised, their arrangement better understood, and an explanation provided for the complex conjugate root pairing.
one real root and a pair of complex conjugate roots.
There is one real root apparent at x = 1 but no indication whatsoever of the whereabouts of the complex conjugate roots.
Consequently, the reflection and transmission coefficients of the new structure become complex conjugates of the old ones.
If the interchanges DPS [left right arrow] DNG and ENG [left right arrow] MNG are made in all the layers except in the half spaces on the two sides of the multilayer structure (which is more realizable), then the reflection coefficients become complex conjugates and the reflected power remains the same.
According to relations (22)-(27) and (34)-(37), if layer l is filled by its dual layer, its dual ABCD matrixes corresponding to TE and TM waves will become equal to the complex conjugates of the former ABCD matrixes.
More generally, if the FFT of one time-domain signal Q is multiplied by the complex conjugate
of the FFT of another signal R, a cross power spectrum results.
alpha]]), N = [+ or -] 1, 2, 3, [alpha] = 1, 2, which may either be real or occur in complex conjugate
This also is a linear relationship, which may be obtained from Equation 3 by post multiplying this expression by its complex conjugate
transpose, to wit (4a) [Mathematical Expression Omitted] (4b) [Mathematical Expression Omitted]
We know that both solutions will be complex, and that the solutions will not be complex conjugates
of each other (because the coefficients are not all real).