Re: Integration

[Bill Allombert <Bill.Allombert@math.u-bordeaux1.fr>]

On Mon, Oct 11, 2010 at 02:54:48PM +0200, Sumaia Saad-Eddin wrote:
> Dear all,
> 
>   here is a simple script I use:
> ----------------------------------------------------
{calF3b(n, y, borne=0)=
   local(res);
   if(borne == 0,
      res = intnum(u = y, [[1], I], 
		   (n*(n-1)*log(u/y)^(n-2)
		    -n*log(u/y)^(n-1)
		    +2*(-n*log(u/y)^(n-1)+log(u/y)^n))*(-cos(u))/u^3
		  )*2/factorial(n),
      res = intnum(u = y, borne, 
		   (n*(n-1)*log(u/y)^(n-2)
		    -n*log(u/y)^(n-1)
		    +2*(-n*log(u/y)^(n-1)+log(u/y)^n))*(-cos(u))/u^3
		  )*2/factorial(n));
   return(res);
}
> Can anyone explain me why these results are so 
> different, or give me a pointer to some litterature?

intnum uses the double-exponential integration method that has strong regularity requirement,
but is often much faster than intnumromb.

The correct integration formula to use is:
intnum(u = y,borne, expr ) = intnum(u = y,[[1],I], expr ) - intnum(borne, [[1],I], expr )

This avoid the exponential growth of the cosinus when the imaginary part get large 
(after the change of variable).

I suggest reading Pascal Molin Ph.D. thesis (Bordeaux, 2010) for more detail.

Cheers,
Bill