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Topic: A bug-looking behavior during integration
Replies: 2   Last Post: Mar 3, 2013 10:58 PM

 Messages: [ Previous | Next ]
 Alexei Boulbitch Posts: 483 Registered: 2/28/08
A bug-looking 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,
L-5326 Contern, LUXEMBOURG

Office phone : +352-2454-2566
Office fax: +352-2454-3566
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e-mail: alexei.boulbitch@iee.lu

Date Subject Author
2/27/13 Alexei Boulbitch
3/2/13 daniel.lichtblau0@gmail.com
3/3/13 Scott Hemphill