paulandrewbird@gmail.com schrieb:   
   >   
   > On Wednesday, 2 July 2014 13:56:36 UTC+1, pauland...@gmail.com  wrote:   
   > > On Tuesday, 1 July 2014 16:12:38 UTC+1, clicl...@freenet.de  wrote:   
   > > > paulandrewbird@gmail.com schrieb:   
   > > > >   
   > > > > Hi, I'm trying to solve the integral:   
   > > > >   
   > > > > integral( exp( a*x^4+4*b*x^3*y+6*c*x^2*y^2+4*d*x*y^3+e*y^4)   
   > > > > ,x=-infty..infty, y=-infty..finty)   
   > > > >   
   > > > > I tried on Wolfram alpha but it was too complicated for it.   
   > > > >   
   > > > > I'm guessing it's going to be in the form 1/P(a,b,c,d,e)^(1/8) where P   
   > > > > is a 4th degree polynomial in the coefficients of the quartic.   
   > > > >   
   > > > > My guess is that P is the discriminant of the polynomial but that is   
   > > > > just a guess.   
   > > > >   
   > > > > Is there any good free software that can solve this? Can Mathematica   
   > > > > or Maple solve this? Can you?   
   > > > >   
   > > >   
   > > > I cannot solve your integral, but find your conjecture must be wrong.   
   > > > The integral remains finite when the quartic discriminant vanishes;   
   > > > taking a=-1, b=0, c=-2, d=0, e=-1 for instance, it numerically evaluates   
   > > > to 1.98670, whereas the quartic discriminant   
   > > >   
   > > > 1/27*(4*(12*a*e - 3*b*d + c^2)^3   
   > > >       - (72*a*c*e - 27*a*d^2 - 27*b^2*e + 9*b*c*d - 2*c^3)^2)   
   > > >   
   > > > vanishes for these coefficients. More generally, since the integrand is   
   > > > stricly positive everywhere, so must be the integral, and its symbolic   
   > > > evaluation must be positive and finite over the entire (a,b,c,d,e)   
   > > > domain where it converges.   
   > > >   
   > >   
   > > Are you sure you included the numbers in the coefficients   
   > (a,4b,6c,4d,e) in your calculation of the discriminant?   
   > >   
   > > A good test would be c=-1/3. Then the discriminant would be zero.   
   > The integral would be:   
   > >   
   > > Int[ exp( -(x^2-y^2)(x^2-y^2) ) ]dxdy = Int[ exp( -u^2 v^2 ) ] with   
   > a change of variables to u=x+y, v=x-y. Which I believe diverges.   
      
   It's possible, even likely, that my computations were inconsistent: I   
   copied the integral from your post, I believe, and the discriminant from   
   some file of mine. I am sure the discriminant does not include your   
   factors (1,4,6,4,1). Sorry for that.   
      
   Anyway, you can write down the most general polynomial of the kind you   
   want, e.g.   
      
     waaaeee*a^3*e^3 + waabdee*a^2*b*d*e^2 + .... + wbbccdd*b^2*c^2*d^2   
      
   numerically evaluate the integral for suitably chosen (a,b,c,d,e) often   
   enough to be able to solve the linear system of equations for (waaaeee,   
   waabdee, ..., wbbccdd), and then check if additional points fit your   
   polynomial too. If not, you have to live with a least-squares   
   approximation or come up with a new idea. The quartic discriminant is of   
   degree six in (a,b,c,d,e), higher or lower degrees may be tried, for   
   example.   
      
   Curious to learn about the results,   
      
   Martin.   
      
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