I have this ugly function :
A[x_]:=((2 x^3 + 9 x^2 + 27 x + 3 Sqrt (3 x^4 + 14 x^3 + 27 x^2)^(1/2))/2)^(1/3)
(notice the fractional exponents at the end, it is indeed ugly) and it so happens that I want to plot the complex solutions
Abs[A[x] + (3x + x^2)/A[x] + x] == Abs[rho A[x] + rho^2 (3 z + z^2)/A[x] + x]
where rho is a cube root of unity (so we can just set it to
(-1 - I Sqrt)/2).
This displays beautifully with ContourPlot and gives a picture of a curve (I actually have to replace x above by x+Iy and both x and y are in a [-3,3] range to see the picture).
In theory, I am supposed to be able to simplify this equation to something a little simpler, i.e. giving something like this :
A[x] Conjugate[x] + Conjugate[(3x+x^2)/A[x]] x + Conjugate[A[x]] (3x + x^2)/A[x] == rho^2 Conjugate[A[x] Conjugate[x] + Conjugate[(3x+x^2)/A[x]] x + Conjugate[A[x]] (3x + x^2)/A[x]]
I have checked the details a thousand times ; getting from the above equation to the latter is just squaring the absolute values, using
|z|^2 = z Conjugate[z] and expanding, it's really trivial math. But when I use ContourPlot to find the solutions to the latter equation, the curve doesn't show up. Any explanations on this?... Anything would be appreciated! From a wild guess to a very detailed attempt or even a suggestion/question.