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I need to graph the function $$f_s(x)=\frac{1+2x^2-3x^2\Xi(x)}{1-x^2},$$ were $$\Xi(x)= \begin{cases} \frac{1}{\sqrt{1-x^2}}\tanh^{-1}(\sqrt{1-x^2}); & 0\leq x<1\\ \frac{1}{\sqrt{x^2-1}}\tan^{-1}(\sqrt{x^2-1}); & x \geq1 \end{cases}.$$

My effort:

P1 = Plot[(1 + 2 x^2)/(1 - x^2) + (3 x^2 ArcTanh[Sqrt[x^2 - 1]])/\!\(\*SuperscriptBox[\((\*SuperscriptBox[\(x\), \(2\)] - 1)\), \({\*FractionBox[\(3\), \(2\)]}\)]\), {x, 1, 14},  PlotTheme -> "Scientific", PlotStyle -> Black,  FrameTicks -> {{{-2, -1.5, -1, 0, 0.5, 1}, 
 None}, {{0, 2, 4, 6, 8, 10, 12, 14}, None}}]

P2 = Plot[(1 + 2 x^2)/(1 - x^2) - (3 x^2 ArcTanh[Sqrt[-x^2 + 1]])/\!\(\*SuperscriptBox[\((\(-\*SuperscriptBox[\(x\), \(2\)]\) + 1)\), \({\*FractionBox[\(3\), \(2\)]}\)]\), {x, 0, 1},  PlotTheme -> "Scientific", PlotStyle -> Black,  FrameTicks -> {{{-2, -1.5, -1, 0, 0.5, 1}, 
 None}, {{0, 2, 4, 6, 8, 10, 12, 14}, None}}]

Show[P1, P2]

I want the graph is equal to enter image description here

or enter image description here

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  • $\begingroup$ Your formula suggests that you should use Tanh, but in your code you use ArcTan. $\endgroup$ – C. E. Jul 12 '17 at 0:10
  • $\begingroup$ it is true. I'm sorry. $\endgroup$ – Dinesh Shankar Jul 12 '17 at 0:16
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Just use $\Xi(x) = \frac{1}{\sqrt{x^2-1}}\tanh(\sqrt{x^2-1})$ for all $x$ instead. Or use $\frac{1}{\sqrt{1-x^2}}\tan(\sqrt{1-x^2})$ for the $x<1$ range instead.

I meant to make this a comment, but since I answered it instead, here is the plot I think you want:

LogLinearPlot[(1+2x^2-3x^2 Tanh[Sqrt[x^2-1]]/Sqrt[x^2-1])/(1-x^2), {x,0,100}]

enter image description here

Update

Looking at the literature, (e.g., Exact solution of the Thomas-Fermi equation for a trapped Bose-Einstein condensate with dipole-dipole interactions by Claudia Eberlein, Stefano Giovanazzi, and Duncan H. J. O’Dell) it seems that the correct function uses ArcTan, and that the formula for $x<1$ is obtained by analytic continuation. So, I think the plot you want is:

LogLinearPlot[(1+2x^2-3x^2 ArcTan[Sqrt[x^2-1]]/Sqrt[x^2-1])/(1-x^2), {x,0.01,100}]

enter image description here

which seems to perfectly match your expected plot.

One more comment. Note:

FullSimplify[
    ComplexExpand[ArcTanh[Sqrt[1-x^2]]/Sqrt[1-x^2]-ArcTan[Sqrt[-1+x^2]]/Sqrt[-1+x^2]],
    x > 0
]

Piecewise[{{0, x != 1}}, Indeterminate]

So, there is no need for a Piecewise definition of $\Xi(x)$. Mathematica can analytically continue both formulas into each other.

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  • $\begingroup$ My function was wrong. I'm so sorry $\endgroup$ – Dinesh Shankar Jul 12 '17 at 0:26
  • $\begingroup$ @user3321 Are you sure it's supposed to be ArcTanh? A brief survey of the literature seems to indicate that it's supposed to be ArcTan. $\endgroup$ – Carl Woll Jul 12 '17 at 0:41
  • $\begingroup$ Now it's right. I'm sorry again $\endgroup$ – Dinesh Shankar Jul 12 '17 at 0:49
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Here is a start:

Ξ[x_] := Piecewise[{{1/Sqrt[1 - x^2]*Tanh[Sqrt[1 - x^2]], 0 <= x < 1}},
  1/Sqrt[x^2 - 1]*Tanh[Sqrt[x^2 - 1]]]

f[x_] := (1 + 2*x^2 - 3 x^2*Ξ[x])/(1 - x^2)

Plot[f[x], {x, 0, 16}, PlotRange -> {Automatic, {-2, 2}}, 
 Frame -> True, AxesStyle

Mathematica graphics

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  • $\begingroup$ My function was wrong. I'm so sorry. $\endgroup$ – Dinesh Shankar Jul 12 '17 at 0:21
  • $\begingroup$ @user3321 No problem, just take from my answer how you define a piecewise function and you can use it with your correct formula. $\endgroup$ – halirutan♦ Jul 12 '17 at 0:31

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