# 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. – bobbym Jun 10 '16 at 4:28
• True. When I mean complex numbers, they are truly complex number, meaning is imaginary part is non-zero. – Sk Sarif Hassan Jun 10 '16 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. – Sk Sarif Hassan Jun 10 '16 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. – Sk Sarif Hassan Jun 10 '16 at 4:04
• Hi; please add that to the original problem. – bobbym Jun 10 '16 at 4:06