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How can I plot the function

F[x_,a_]:=Integrate[Abs[x - z]^(-1-2a), {z, -Infinity, -1}] + Integrate[Abs[x - z]^(-1-2a), {z, 1, Infinity}]

for x in (-1,1) and a in (0,1)? I have been using Wolfram Cloud to do Plot[F[x,1], {x,-1,1}], but I get that the computational time is exceeded So my questions are:

  • What does the plot look like? Is this an error or just a long computation?
  • If I wrote Plot[F[x,a], {x,-1,1}, {a,0,1}] would it work to plot in the same picture the function F[x] for the various values of a? What if I just want to plot some values of a, for example a = 1/4, 1/3, 1/2, 2/3, 3/4, 1?
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2 Answers 2

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Using some assumptions and an immediate assignment gives an analytic integral, which is easy to plot:

F[x_, a_] = Assuming[-1 <= x <= 1 && a > 0,
  Integrate[Abs[x - z]^(-1 - 2 a), {z, -Infinity, -1}] + 
  Integrate[Abs[x - z]^(-1 - 2 a), {z, 1, Infinity}]]

(*    (1 - x)^(-2 a)/(2 a) + (1 + x)^(-2 a)/(2 a)    *)

The plot looks exactly like the one of @Andreas' answer.

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  • $\begingroup$ Thanks! Can i ask you too if it's possible to plot it in 1-d, with lines of different colors to represent the various values of a? $\endgroup$
    – Jun
    Commented Apr 16, 2021 at 20:21
  • $\begingroup$ Plot[Evaluate@Table[F[x, a], {a, {1/4, 1/3, 1/2, 2/3, 3/4, 1}}], {x, -1, 1}] $\endgroup$
    – Roman
    Commented Apr 16, 2021 at 20:22
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You can do numerical integration and a 3D plot like:

Plot3D[NIntegrate[Abs[x - z]^(-1 - 2 a), {z, -Infinity, -1}] + 
NIntegrate[Abs[x - z]^(-1 - 2 a), {z, 1, Infinity}], {x, -1, 1},    {a,0,1}]

enter image description here

a 1-d plot could look like

Plot[Evaluate[
 Table[((1 - x)^(-2*a) + (1 + x)^(-2*a))/(2*a), {a, 0.1, 1, 0.2}]], 
{x, -1, 1}]

enter image description here

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  • $\begingroup$ Thank you. I still get the output on the computational time exceeded though. Could you please add a plot to your answer? $\endgroup$
    – Jun
    Commented Apr 16, 2021 at 20:03
  • $\begingroup$ Thank you. Is it also possible to plot it in 1-d, with lines of different colors to represent the various values of a? $\endgroup$
    – Jun
    Commented Apr 16, 2021 at 20:12
  • $\begingroup$ with @Roman s result Plot[Evaluate[ Table[((1 - x)^(-2*a) + (1 + x)^(-2*a))/(2*a), {a, 0.1, 1, 0.2}]], {x, -1, 1}] $\endgroup$
    – Andreas
    Commented Apr 16, 2021 at 20:27
  • $\begingroup$ Thanks. Could you add a picture of this too? $\endgroup$
    – Jun
    Commented Apr 16, 2021 at 20:33
  • $\begingroup$ Thank you so much! This is a big help. Is it possible to also have a legend of which colors correspond to which values of a? $\endgroup$
    – Jun
    Commented Apr 16, 2021 at 21:03

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