Re: Integration Methods in PARI

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

On Fri, Mar 16, 2018 at 11:24:52PM +0100, Bill Allombert wrote:
> 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!)

A proof of this is there:
https://math.stackexchange.com/questions/97558/prove-fourier-series-of-e-cos-x-sin-sin-x-is-sum-n-0-infty-frac

> 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