Re: Integration Methods in PARI

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

On Fri, Mar 16, 2018 at 09:13:50PM +0100, kevin lucas wrote:
> I made a mistake copying the integral from paper, it should have been
> intnum(x=0, +oo, x*exp(cos(x))*sin(sin(x))/(x^2+1))
> Any help or references, PARI-specific or otherwise, for integrating such
> oscillating integrals are welcome. I apologize for the mistake.

Assuming the following (I did not attempt to prove it):

exp(cos(x))*sin(sin(x)) = sum(n=1,oo,sin(n*x)/n!)

then set

si(n)=intnum(x=0, [oo,-n*I] , x*sin(n*x)/(x^2+1))

then you integral should be:

suminf(n=1,si(n)/n!)

which is about 0.698482642717884272267230358497712444

Cheers,
Bill