Reduce[Abs[1/(1 + I/Sqrt[α])] < 0.5, {α}]
It is taking long time to run. Can any one help me to get the reduced condition in terms of $\alpha$ or its modulus. Note that $\alpha$ is a complex number.
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Sign up to join this communityReduce[Abs[1/(1 + I/Sqrt[α])] < 0.5, {α}]
It is taking long time to run. Can any one help me to get the reduced condition in terms of $\alpha$ or its modulus. Note that $\alpha$ is a complex number.
First of all you shouldn't (if possible) use approximate numbers working with symbolic functionality like a very sophisticated function Reduce
.
Before seeking the set of your interest try to envisage the region:
RegionPlot[ Abs[1/(1 + I/Sqrt[x + I y])] < 1/2,
{x, -1.1, 0.5}, {y, -0.6, 1.0}]
Now we can see what we are to find, i.e. dependence of the boundary of this set, i.e. we should find a few functions yielding y
as a function of x
on the boundary.
We can do it with Reduce
if α
will be rewritten as x+ I y
:
Reduce[ Abs[1/(1 + I/Sqrt[x + I y])] == 1/2 && x ∈ Reals && y ∈ Reals, {x, y}]
(x == -1 && y == 0) || (-1 < x < -(1/9) && y == Root[1 + 4 x - 42 x^2 + 36 x^3 + 81 x^4 + (-46 + 36 x + 162 x^2) #1^2 + 81 #1^4 &, 2]) || (x == -(1/9) && y == 4/(3 Sqrt[3])) || (-(1/9) < x <= 1/3 && ( y == Root[1 + 4 x - 42 x^2 + 36 x^3 + 81 x^4 + (-46 + 36 x + 162 x^2) #1^2 + 81 #1^4 &, 2] || y == Root[1 + 4 x - 42 x^2 + 36 x^3 + 81 x^4 + (-46 + 36 x + 162 x^2) #1^2 + 81 #1^4 &, 4]))|| (1/3 < x < 7/18 && ( y == Root[1 + 4 x - 42 x^2 + 36 x^3 + 81 x^4 + (-46 + 36 x + 162 x^2) #1^2 + 81 #1^4 &, 3] || y == Root[1 + 4 x - 42 x^2 + 36 x^3 + 81 x^4 + (-46 + 36 x + 162 x^2) #1^2 + 81 #1^4 &,4])) || (x == 7/18 && y == Sqrt[5/3]/6)
To figure out why the solution looks slightly involved see e.g.
Plot[ ReIm /@ Table[ Root[ 1 + 4 x - 42 x^2 + 36 x^3 +
81 x^4 + (-46 + 36 x + 162 x^2) #1^2 + 81 #1^4 &, k],
{k, {2, 3, 4}}] // Flatten,
{x, -1, 1}, Evaluated -> True, Exclusions -> {-(1/9)},
AspectRatio -> Automatic, PlotStyle -> Thick]
There are three roots involved to describe the boundary, this also clarifies why it was difficult to find this set directly using α
as a complex unknown.
If we plot all the roots of the underlying polynomial then the curves will form more symmetric pattern:
Plot[ ReIm /@ Table[ Root[ 1 + 4 x - 42 x^2 + 36 x^3 + 81 x^4
+ (-46 + 36 x + 162 x^2) #1^2 + 81 #1^4 &, k], {k, 4}] // Flatten,
{x, -1, 1}, Evaluated -> True, Exclusions -> {-(1/9)},
AspectRatio -> Automatic, PlotStyle -> Thick]
There were interesting posts discussing how roots of polynomials are numbered.