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A bug-looking behavior during integration


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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