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I have defined this two quantities:

f = (-b + Sqrt[b^2 + 4 a])/(2 a)
g = (-b - Sqrt[b^2 + 4 a])/(2 a)

and I want to solve this system in function of $a$, for $f,g$ complex: $$\begin{cases} |f(a,b)|>1 \\ |g(a,b)|>1 \\ Im(a)=Im(b)=0 \end{cases}$$

So I use:

Reduce[Abs[f] > 1 && Abs[g] > 1 && Im[a] == 0 &&  Im[b] == 0, a, Complexes]

getting the solution I seached.

total solution

Now, I want to split the regions where the square root in the equation of $f,g$ is positive or negative. If I use:

Reduce[Abs[f] > 1 && Abs[g] > 1 && Im[a] == 0 &&   Im[b] == 0 && (Re[b^2 + 4 a] >= 0 || 
Re[b^2 + 4 a] < 0) , a, Complexes]

I got the same result as before, and this is true. My ploblem is that if a solve separately

Reduce[Abs[f] > 1 && Abs[g] > 1 && Im[a] == 0 && Im[b] == 0 &&   Re[b^2 + 4 a] >= 0 , a, Complexes]
Reduce[Abs[f] > 1 && Abs[g] > 1 && Im[a] == 0 && Im[b] == 0 &&   Re[b^2 + 4 a] < 0 , a, Complexes]

and I plot the results using RegionPlot a small area disapears.

solutions missing

How can I solve this problem?

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  • $\begingroup$ @Öskå Thank you for your answer. I added MaxRecursion to the options of RegionPlot and I fixed the problem. Can you explain me why without it RegionPlot does not work? $\endgroup$
    – Marco
    Oct 8 '14 at 14:22
  • $\begingroup$ See MaxRecursion for the details :) $\endgroup$
    – Öskå
    Oct 8 '14 at 14:23
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You need to increase the MaxRecursion:

f = (-b + Sqrt[b^2 + 4 a])/(2 a);
g = (-b - Sqrt[b^2 + 4 a])/(2 a);
RegionPlot[{
  Reduce[Abs[f] > 1 && Abs[g] > 1 && Im[a] == 0 && 
             Im[b] == 0 && Re[b^2 + 4 a] >= 0, a, Complexes],
  Reduce[Abs[f] > 1 && Abs[g] > 1 && Im[a] == 0 && 
             Im[b] == 0 && Re[b^2 + 4 a] < 0, a, Complexes]}, {a, -2, 2}, {b, -3, 3}, 
 MaxRecursion -> 7, Evaluated -> True]

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  • $\begingroup$ What is the option Evaluated -> True for RegionPlot? I was not able to find it in the documentation. $\endgroup$ Jul 20 '17 at 18:53
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    $\begingroup$ @TaikiBessho, Try it without, and you will see the difference :) You can always look for the Evaluated documentation. $\endgroup$
    – Öskå
    Jul 20 '17 at 20:28

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