How to find this convolution?

Question

How can I use Mathematica to find the convolution $f*f$ for

f[t_]:=Piecewise[
        {{t*(Pi - 4*t + t^2), Inequality[0, Less, t, LessEqual, 1]}, 
         {(-t)*(2 + t^2 - 4*Sqrt[-1 + t^2] - 2*ArcCsc[t] + 2*ArcTan[Sqrt[-1 + t^2]]), 
           1 < t < Sqrt[2]
        }}, 0]

I would need a symbolic expression for the convolution. Thanks for any help.

Answer 1

Hard to do it analytically. Tried convolution theorem also. ForuierTransform had hard time with it as well as Integrate. So, here is a numerical solution.

The support needed is really only from $0$ to $2 \sqrt(2)$ since your function exist over $0$ to $\sqrt(2)$ but I integrated it over little larger range for the plot to look better.

Hence

  f[t_?NumericQ] := 
  Piecewise[{{t*(Pi - 4*t + t^2), 
     Inequality[0, Less, t, LessEqual, 
      1]}, {(-t)*(2 + t^2 - 4*Sqrt[-1 + t^2] - 2*ArcCsc[t] + 
        2*ArcTan[Sqrt[-1 + t^2]]), 1 < t < Sqrt[2]}}, 0];
g[t_?NumericQ] := 
 NIntegrate[f[tao] f[t - tao], {tao, -Infinity, Infinity}]
data = Table[{t, g[t]}, {t, -0.5, 6, .01}];

Show[ListLinePlot[data, PlotStyle -> Red, PlotRange -> All], 
 Plot[f[t], {t, -.5, 6}, PlotRange -> All, Exclusions -> None]]

The red plot is the convolution and the blue curve is $f(t)$