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I have to plot this function:

Q[ω_] := ArcTan[1/ω]/(1 + 1/2*Log[(1 + ω^2)/ω^2])

And I did it through

Plot[Q[ω], {ω, 0, 3},
  PlotRange -> {0, 1}, 
  PlotStyle -> {Black}]

In a very simple way.

The output is the following:

enter image description here

But here is the problem: the function, as $\omega \to 0$ must be zero.

Why the plot doesn't show that behaviour?

The convergence to zero is really slow (to say: for $\omega \to 10^{-309}$ I just obtained $0.0022$). Is there a way to make the plot to start from zero?

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2 Answers 2

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The plot is showing this behavior. Change the PlotRange to see it more clearly.

Plot[Q[ω], {ω, 0, 3}, 
 PlotRange -> {{0, 0.05}, Automatic}, PlotStyle -> {Black}]

Better still is to check the limit of the function as ω -> 0

In[6]:= Limit[Q[ω], ω -> 0]

Out[6]= 0
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  • $\begingroup$ Ah, teak you! I suspected it was because of the very slow convergence to zero... Here it is! Thank you so much! $\endgroup$
    – Enrico M.
    Commented May 17, 2017 at 9:43
  • $\begingroup$ I don't think this solves the problem. I guess one can do it with appropriate reparametrization and plotting the graph with PolarPlot. $\endgroup$
    – Artes
    Commented May 17, 2017 at 10:11
  • $\begingroup$ @Artes Actually it did. $\endgroup$
    – Enrico M.
    Commented May 17, 2017 at 12:02
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    $\begingroup$ @HenryTuring The problem was to make an appropriate plot, when I draw it with PlotRange -> {{0, 0.05}, Automatic} values of Q start roughly from 0.1, and when I take PlotRange -> {{0, 0.001}, Automatic} I encounter the same issue, i.e. it does not start from {0, 0}. $\endgroup$
    – Artes
    Commented May 17, 2017 at 12:11
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If you plot slightly into the negative then limit the PlotRange to positive values you get this:

Plot[Q[ω], {ω, -0.01, 3}
 , PlotRange -> {{0, 3}, {0, Full}}
 , PlotStyle -> {Black}
 , AxesOrigin -> {1, 0}
]

enter image description here

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  • $\begingroup$ No need to go negative; PlotRange -> {Automatic, {0, Full}} actually suffices: Plot[Q[ω], {ω, 0, 3}, AxesStyle -> Opacity[1/2], PlotRange -> {Automatic, {0, Full}}, PlotStyle -> {Black}] $\endgroup$
    – J. M.'s missing motivation ♦
    Commented May 17, 2017 at 12:31
  • $\begingroup$ @J.M. In version 10.1.0 under Windows I do in fact need to, and with your code the line doesn't make it all the way to zero. $\endgroup$
    – Mr.Wizard
    Commented May 17, 2017 at 12:33
  • $\begingroup$ Huh. I guess it's something they did in 11, then... $\endgroup$
    – J. M.'s missing motivation ♦
    Commented May 17, 2017 at 12:38

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