Vtyjkyuk

In my recent work, I need to the details of the computation of the following multiple improper integral:
$$iint_[0,1]^2e^-pi x^2y^2dxdy-iint_[1,infty)^2e^-pi x^2y^2dxdy.$$
As you see, the first one is a proper integral and the second one is improper integral. It is easy to know the convergence of the second one. At the beginning, I just use Wolfram mathematica 9.0 to get the finally result:
$$frac14(gamma+log(4pi)),$$
where $gamma$ is a Euler-Gamma constant.
But now I need to give the details of the computation.
The wolfram mathematica tells us that:
$$iint_[0,1]^2e^-pi x^2y^2dxdy=HypergeometricPFQ[(1/2,1/2);(3/2,3/2);-pi];$$
$$iint_[1,infty)^2e^-pi x^2y^2dxdy
=HypergeometricPFQ[(1/2,1/2);(3/2,3/2)-frac14(gamma+log(4pi)).$$
I do not know how to get the hyperbolic geometric function and how to cancel it.
Any help and hints will welcome. Thanks a lot.

From the definition of Error Function:
$$
beginalign
I_1&=+iint_[0,1]^2e^-pi,x^2 y^2,dx dy,=,int_0^1dx,int_0^1+,e^-pi x^2 y^2,dy,=,int_0^1left[fractexterfleft(sqrtpi,x yright)2xright]_y=0^y=1,dx \[2mm]
&=int_0^1fractexterfleft(sqrtpi,xright)2x,dxqquadcolorbluesmall u=texterfleft(sqrtpi,xright),Rightarrow,du=2e^-pi,x^2;,,dv=fracdx2x,Rightarrow,v=fraclogx2\[2mm]
&=left[left(,texterfleft(sqrtpi,xright)right)fraclogx2right]_0^1,-,int_0^1e^-pi,x^2,logx,dx,=-int_0^1e^-pi,x^2,logx,dxqquadqquadqquadcolorblue1 \[6mm]
I_2&=-iint_[1,infty)^2e^-pi,x^2 y^2,dx dy,=,int_1^inftydx,int_1^infty-,e^-pi x^2 y^2,dy,=,int_1^inftyleft[fractexterfleft(sqrtpi,x yright)-12xright]_y=infty^y=1,dx \[2mm]
&=int_1^inftyfractexterfleft(sqrtpi,xright)-12x,dxqquadcolorbluesmall u=texterfleft(sqrtpi,xright)-1,Rightarrow,du=2e^-pi,x^2;,,dv=fracdx2x,Rightarrow,v=fraclogx2\[2mm]
&=left[left(,texterfleft(sqrtpi,xright)-1right)fraclogx2right]_1^infty,-,int_1^inftye^-pi,x^2,logx,dx,=-int_1^inftye^-pi,x^2,logx,dxqquadquadcolorblue2 \[6mm]
colorredI&=I_1+I_2=-int_0^inftye^-pi,x^2,logx,dx=-int_0^inftye^-,x^2,logfracxsqrtpi,,fracdxsqrtpiqquadcolorbluesmall x=fractsqrtpi\[2mm]
&=-frac1sqrtpiint_0^inftye^-,x^2,logx,dx+fraclogpi2sqrtpiint_0^inftye^-,x^2,dx=small fracgamma+log44+fraclogpi4=colorredfracgamma+log4pi4quadcolorblue3
endalign
$$

$,colorblue1,$ Two times L'Hopital's Rule, starting by $,displaystylesmall lim_xrightarrow 0left[texterf(x).logxright]=lim_xrightarrow 0,fraclogx1/texterf(x),$

$,colorblue2,$ Four times L'Hopital's Rule, starting by $,displaystylesmall lim_xrightarrow inftyleft[texterfc(x).logxright]=lim_xrightarrow infty,fractexterfc(x)1/logx,$

$,colorblue3,$ Integral of Gamma Conatant.