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Dirk Laurie on Mon, 19 Mar 2018 14:48:27 +0100
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Re: Integration Methods in PARI
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- To: pari-users@pari.math.u-bordeaux.fr
- Subject: Re: Integration Methods in PARI
- From: Dirk Laurie <dirk.laurie@gmail.com>
- Date: Mon, 19 Mar 2018 15:48:17 +0200
- Delivery-date: Mon, 19 Mar 2018 14:48:27 +0100
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- In-reply-to: <20180316222452.GC29248@yellowpig>
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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)