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Re: A bug-looking behavior during integration


On Wednesday, February 27, 2013 10:42:23 PM UTC-6, Alexei Boulbitch wrote:
> Dear community,
>  
> I would like to report a strange behaviour, that looks most of all like a bug.
> 
> The integral of the UnitStep of some function
>  
> Integrate[
>  UnitStep[Sqrt[Sqrt[x^2 + y^2] + x] -
>    0.7 Sqrt[x^2 + y^2]], {x, -Infinity, Infinity}, {y, -Infinity,Infinity}]
> 
> 0
> 
> returns zero as you see, though the function under the integral is 1 within some domain in the form of a cardioid, and zero outside of it. To make it sure evaluate this:
> 
> Plot3D[UnitStep[
> 
>   Sqrt[Sqrt[x^2 + y^2] + x] - 0.7 Sqrt[x^2 + y^2]], {x, -3, 5}, {y, -3, 5}]
> 
> The integral must be positive, therefore.
> 
> Taking the numerical value of the same integral one finds a finite positive value:
> 
> NIntegrate[
>  UnitStep[Sqrt[Sqrt[x^2 + y^2] + x] -
>    0.7 Sqrt[x^2 + y^2]], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]
> 
> 19.6268
> 
> Is it indeed a bug?
> 
> 
> Regards, Alexei

Yes, it looks like this runs into some fragility in the UnitStep handling code. To work around that area I'd suggest using a Piecewise equivalent.

In[2]:= Integrate[                                                              
         Piecewise[{{1, Sqrt[Sqrt[x^2 + y^2] + x] - .7*Sqrt[x^2 + y^2] >= 0}},  
           0], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]              

Out[2]= 19.6268

Daniel Lichtblau
Wolfram Research




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