Re: Integration Methods in PARI

[kevin lucas <lucaskevin296@gmail.com>]

I found something similar in Gradshteyn & Ryzhik (7th ed., pg. 512, Formula 3.973 (2), here)

 but cannot prove this closed form. How was it obtained?

On Mon, Mar 19, 2018 at 2:48 PM, Dirk Laurie <dirk.laurie@gmail.com> wrote:

2018-03-17 0:24 GMT+02:00 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

Pi/2*(exp(exp(-1))-1)

(thanks to André Weideman of the University of Stellenbosch)