Unexpected result from the integration over 2D gauss function

From: Andrey Kormilitzin <andreyk1_at_post.tau.ac.il>
Date: Sun, 22 Jul 2007 13:50:17 +0300

Dear Rooter,

I need to perform a very simple 2D integration over 2D gaussian function (x,y):

\int_0^inf \int_0^inf dx dy 4*pi^2*x*y*exp(- (x^2+y^2)/S^2 )/pi^2/S^2

The correct answer is 1, but I obtain another result (25.6846). Actually, there is no reason for that. I checked the 1D integration over the 1D gaussian function and it works well. I want to overcome the change of variables, since my real integrand looks more complicated as a function of 'x' and 'y'.

The code is very simple:

//**************************************************************************

#define S_p 1.0
#define PI TMath::Pi()
#define eps 1e-8

double proton_wave_function(double x, double y) {// normalized 2D gaussian

     return TMath::Exp( - (x*x + y*y)/S_p/S_p )/PI/PI/S_p/S_p; }

double integrand_PP(double* var, double* par) {

	double x		= var[0];
	double y		= var[1];

	double t1		= ((1-x)/x);
	double J_t1		= 2*PI*(1/x/x)*t1;

	double t2		= ((1-y)/y);
	double J_t2		= 2*PI*(1/y/y)*t2;

	return J_t1*J_t2*proton_wave_function(t1, t2);

}

double sigma_tot_PP()
{
double limit_d[2] = {eps,eps} ,limit_u[2] = {1-eps,1-eps}, ifail=0.0, relerr=0.0, nfnevl=0.0;
int min_p = 10, as1, as2;

TF1* soft = new TF1("soft_int",integrand_PP,1);

return (soft->IntegralMultiple(2, limit_d, limit_u, min_p, 5e4, 1e-20, relerr, as1, as2));

}

//**************************************************************************

• I made a change of variables (x,y)->(t1,t2) in order to change the limits of integration from 0->inf to 0->1. J_t1, J_t2 correspond to jacobian of the transformation.

Is that problem can be solved without the change of variables x^2 + y^2 -> r^2 ?

Sincerely,

                      Andrey K.

   *************************************************************
   * Andrey Kormilitzin    *                                   *
   *                       *                                   *
   * Department  of        *                                   *
   * Particle Physics      *                                   *
   * School of Physics     *  Tel: ++ 972- 3 - 640 7954 (o)    *
   * Tel Aviv University   *                                   *
   * Ramat Aviv,Tel Aviv   *  E-mail: andreyk1_at_post.tau.ac.il  *
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