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Problem with integrate and assumptions

Posted 15 days ago
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Dear all,

I am struggling to understand why the following integration:

Integrate[t*BesselJ[0,k*r]*(-HeavisideTheta[t-r/a]+HeavisideTheta[t-r/b])/r^2,{r,0,Infinity},Assumptions->{k>=0,a>0,b>0,a>b,t>=0}]

Is not giving the same result as (notice that only the assumption about t>0 or t>=0 has changed):

Integrate[t*BesselJ[0,k*r]*(-HeavisideTheta[t-r/a]+HeavisideTheta[t-r/b])/r^2,{r,0,Infinity},Assumptions->{k>=0,a>0,b>0,a>b,t>0}]

I am expecting the result to be non-zero, because this integration arise from calculating the Fourier transform of a non-zero function in 2D.

Do you have any ideas why the calculation gives two different results?

I am using Mathematica Version 12.2.0.0 on Mac.

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3 Replies

Hello Pierre,

with my stone-age-old Version 7 both integrals give zero.

Perhaps you should check this numerically. Your integrand is, let's say, somewhat special (for a > b I wrote a = b + x and x >0 ). But I am quite sure that there are combinations of the parameters that the integral is different from zero.

Manipulate[
 Plot[Evaluate[ t*BesselJ[0, k*r]*(-HeavisideTheta[t - r/(b + x)] + HeavisideTheta[t - r/b])/     r^2], 
{r, 0, 5}, PlotRange -> {-.2, .2}],
 {t, 0, 5}, {x, .01, 5}, {k, 0, 5}, {b, .1, 5}]

Dear Hans,

Thank you a lot for your answer. I will try what you said. I am also quite sure this integral is non-zero except for negative t.

I will file a bug report for this.

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