# Stuck in Reduce and Findinstance function for the following problem

    FindInstance[(Abs[(
2 β)/(β +
Sqrt[-4 α + (β - γ)^2] + γ)] +
Abs[(4 α -
2 β (β +
Sqrt[-4 α + (β - γ)^2] - γ))/(\
β +
Sqrt[-4 α + (β - γ)^2] + γ)^2]) <
1, {α, β, γ}, 10]


It was running for a long time. Unfortunately no output came. Can anyone help to get it done? Also I was trying to reduce the inequality in terms of $\alpha$, $\beta$ and $\gamma$ but again same problem happened. Help to get it done, please. Note that $\alpha$, $\beta$ and $\gamma$ are complex numbers.

• Integers are complex numbers with the coefficient of the imaginary part being 0, so both answers below are finding valid solutions. Jun 10, 2016 at 4:28
• True. When I mean complex numbers, they are truly complex number, meaning is imaginary part is non-zero. Jun 10, 2016 at 4:40

The problem seems to be with the Abs[ ] function. Since you have the sum of two Abs[ ] terms, and since you are only looking for specific examples, you can rewrite your FindInstance as:

FindInstance[{-1/2 < (2 \[Beta])/(\[Beta] +
Sqrt[-4 \[Alpha] + (\[Beta] - \[Gamma])^2] + \[Gamma]) < 1/2,
-1/2 < (4 \[Alpha] - 2 \[Beta] (\[Beta] +
Sqrt[-4 \[Alpha] + (\[Beta] - \[Gamma])^2] - \[Gamma]))/
(\[Beta] + Sqrt[-4 \[Alpha] + (\[Beta] - \[Gamma])^2] + \[Gamma])^2 < 1/2},
{\[Alpha], \[Beta], \[Gamma]}]


This returns {[Alpha] -> -7, [Beta] -> -1, [Gamma] -> 2}, which can be verified to fulfill the original equation as well.

• Can we reduce the inequality? Actually the $\alpha$, $\beta$ and $\gamma$ are all complex numbers. Jun 10, 2016 at 3:59

FindInstance[{10 > \[Alpha] > 0, 10 > \[Beta] > 0,
10 > \[Gamma] >
0, (Abs[(2 \[Beta])/(\[Beta] +
Sqrt[-4 \[Alpha] + (\[Beta] - \[Gamma])^2] + \[Gamma])] +
Abs[(4 \[Alpha] -
2 \[Beta] (\[Beta] +
Sqrt[-4 \[Alpha] + (\[Beta] - \[Gamma])^2] - \[Gamma]))/(\
\[Beta] + Sqrt[-4 \[Alpha] + (\[Beta] - \[Gamma])^2]  +\[Gamma])^2]) <1}, {\[Alpha], \[Beta], \[Gamma]}, Integers, 10]


I upvoted bill s' solution but some answers are better than nothing.

• Actually the α, β and γ are all complex numbers. What is more important to get complex instances rather than real numbers. Also if we could make the reduced inequality, that would be sufficient. Jun 10, 2016 at 4:04
• Hi; please add that to the original problem. Jun 10, 2016 at 4:06