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.   
   >   
   >   
   >   
   > Martin.   
      
   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.   
      
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