I want to check when the expression
$$ \sqrt{\sqrt{\lambda^{2}\left(2E+\lambda^{2}\right)}+E+\lambda^{2}} $$
is real, when it is purely imaginary and when it is complex (with imaginary part not zero).
Also I would like to know the sign of the imaginary part in the case it is non-zero. To check the last part, after some steps, I am using Reduce
for inequalities like $\mathrm{Im}\left(\sqrt{\lambda^{2}\left(2E+\lambda^{2}\right)}\right)>0 $ but this is taking a lot of time on Mathematica. Is there a way to get the kind of results that I want more quickly?
The variables are real and the code I ran is
Reduce[ Im[ Sqrt[ e + λ^2 -Sqrt[ λ^2 (2 e + λ^2)]]] > 0, {e, λ}]
Reduce[Im[Sqrt[e + x^2 + Sqrt[(x^2) (2*e + x^2)]]] == 0, x, Reals]
for real vars $\endgroup$ – Dr. belisarius Jul 16 '13 at 17:55