Unexpected integration result?

Questions
,
1

I am getting a strange integration result: the integral of the absolute value of a non-zero function gives me 0 as an output. There is a MWE in the attached notebook.
Can give me a hint about how to solve the issue?

(I am using Mathematica 12.0.)

test_integration.png
test_integration.png1038×288 17.8 KB

test_integration.nb (3.21 KB)

2

The result is clearly wrong. However, it is not reasonable to expect Mathematica to solve a trascendental equation with two parameters to find where f=0. It can find a closed form when the parameters are numbers:

With[{\[Mu] = 2, \[Sigma] = 1},
 Integrate[
  Abs[f[x, \[Mu], \[Sigma]]], {x, -\[Infinity], \[Infinity]}]]
3

Indeed, the function to integrate is not among the easiest. However, I would have expected Mathematica to realise the problem cannot be solved instead of giving a wrong answer.

Moreover, elevating the function to the power 2 outputs what seems like a reasonable output. To which extent should I trust this result?

Assuming[{m > 0, s > 0}, 
 Integrate[f[x,m,s]^2, {x, -\[Infinity], \[Infinity]}]]
4

The square of f has a closed-form primitive. You can check against that.

5

The output of the primitive of the absolute value of f, namely “0”, is also a closed form (even if not the correct one). How can I be sure that the second result is sound if the other one failed?

6

This is an interesting problem. Some remarks:

fm[x_, m_, s_] := Exp[-(x - m)^2/(2 s^2)]/s
fp[x_, m_, s_] := Exp[-(x + m)^2/(2 s^2)]/s
f2[x_, m_, s_] := Exp[-x^2/(2 (m^2 + s^2))]/Sqrt[m^2 + s^2]

Table[Plot[{fm[x, i, j], fp[x, i, j], f2[x, i, j]},
  {x, -10, 10}, PlotRange -> All],
 {i, 1, 3}, {j, 1, 3}]

ff[x_, m_, s_] := fm[x, m, s] + fp[x, m, s] - 2 f2[x, m, s]

Manipulate[
 Plot[{ff[x, i, j], Abs[ff[x, i, j]]},
  {x, -5, 5}, PlotRange -> {-1, 1}],
 {i, 1, 3}, {j, 1, 3}]

Integrate[Abs[ff[x, 1, 1]], {x, -Infinity, Infinity}] // N
NIntegrate[Abs[ff[x, 1, 1]], {x, -100, 100}]
Integrate[Abs[ff[x, 2, 1]], {x, -Infinity, Infinity}] // N
NIntegrate[Abs[ff[x, 2, 1]], {x, -50, 50}]


vtab = Table[
   {i, j, NIntegrate[Abs[ff[x, i, j]], {x, -50, 50}]},
   {i, 1, 3, .1}, {j, 1, 3, .1}];
ListPlot3D[Flatten[vtab, 1]]


tt = Table[{j, NIntegrate[Abs[ff[x, j, 1]], {x, -50, 50}]}, {j, 2, 20}];
ListLinePlot[tt]

It may be that the integral is bounded.

7

As for the primitive of Abs[ff[x,m,s]], I get back the input, not 0.

The square of ff is a simple sum of Gaussians, there is no problem in finding a primitive in terms of the error function Erf, and in calculating the limits at infinity.

8

Yes, without restriction over the parameters, the integral isn’t computed. Adding the assumptions leads to the aforementioned strange result.