Re: A bug-looking behavior during integration

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• Subject: [mg129970] Re: A bug-looking behavior during integration
• From: Alex Krasnov <akrasnov at eecs.berkeley.edu>
• Date: Thu, 28 Feb 2013 21:27:02 -0500 (EST)
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```This does appear to be a bug. Two other examples (Mathematica 8.0.4):

In:	Integrate[UnitStep[(1+Cos[ArcTan[x, y]])-Sqrt[x^2+y^2]], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]
Out:	0

In:	Integrate[UnitStep[Cos[3*ArcTan[x, y]]-Sqrt[x^2+y^2]], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]
Out:	0

Alex

On Wed, 27 Feb 2013, 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
>
>
> Alexei BOULBITCH, Dr., habil.
> IEE S.A.
> ZAE Weiergewan,
> 11, rue Edmond Reuter,
> L-5326 Contern, LUXEMBOURG
>
> Office phone :  +352-2454-2566
> Office fax:       +352-2454-3566
> mobile phone:  +49 151 52 40 66 44
>
> e-mail: alexei.boulbitch at iee.lu
>
>
>
>

```

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