
A buglooking behavior during integration
Posted:
Feb 27, 2013 11:42 PM


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, L5326 Contern, LUXEMBOURG
Office phone : +35224542566 Office fax: +35224543566 mobile phone: +49 151 52 40 66 44
email: alexei.boulbitch@iee.lu

