Re: A bug-looking behavior during integration

*To*: mathgroup at smc.vnet.net*Subject*: [mg129971] Re: A bug-looking behavior during integration*From*: daniel.lichtblau0 at gmail.com*Date*: Thu, 28 Feb 2013 21:27:22 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <kgmn7f$8gj$1@smc.vnet.net>

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