The complex conjugate roots do not correspond to the locations of either turning point or the PoI shown in Figure 1 and have often been misplaced by well-meaning but ingenuous authors like Stroud (1986, see Programme 2, Theory of Equations, where he [incorrectly] places the complex conjugate roots in the Cartesian plane at one of the turning points of a cubic equation).
For the specific example considered here, Figures 6 and 7 show one real root and a pair of complex conjugate roots located respectively at (G, H) or (Re(x), Im(x)) = (1, 0), (-2, 1) and (-2, -1) which accords fully with the solution for y = [x.
The manner in which the principal planes of symmetry of surfaces A and B intersect each other at right angles through the umbilical point helps explain why the complex conjugate roots have to occur in pairs equidistant from the G-axis; they can now be visualised occurring behind and in front of the original Cartesian x-y plane.
Figure 8(a) shows another instance where the complex conjugate roots are not evident in the Cartesian plane.
There is one real root at x = -1, and a complex conjugate pair at x = 1 [+ or -] 2i.
Consequently, the overall ABCD matrixes of the dual multilayer planar structure (as the product of the individual line sections) corresponding to TE and TM waves will become equal to the complex conjugates of those of the original structure.
Considering the fact that the elements of ABCD matrixes of the two dual structures are complex conjugates of each other and that [[eta].
If the structure is backed by a perfect electric conductor and the media interchanges DPS [left right arrow] DNG, and ENG [left right arrow] MNG are made in the layers, then the reflection coefficients of the two dual structures become complex conjugates of each other, and the reflected powers are equal.