Re: Problème avec intnum

[Damian Rössler <damian.rossler@maths.ox.ac.uk>]

Thank you for the explanations. I understand now. Regards, Damian Rössler

> On 12 Jun 2023, at 9:50 pm, Bill Allombert <Bill.Allombert@math.u-bordeaux.fr> wrote:
> 
> On Mon, Jun 12, 2023 at 09:09:50PM +0100, Damian Rossler wrote:
>> Dear Bill, thank you for this. What do you mean by « an asymptotic singularity at I »? 
>> The integral is along the real line, so it never meets any singularity of the function (in particular, log(x+I) is always well-defined). 
> 
> I meant that:
> 
>>> PARI uses the double exponential method that does a change of variable. Unfortunately this causes the singularity
>>> at I to get closer and closer to the integration path when N goes to infinity.
> 
> As you say, there are no singularity. However if you integrate over [-x,x],
> after variable change, the singularity became closer and closer to the
> integration interval when x goes to infinity. This is explained for example in
> Pascal Molin thesis.
> 
> If you integrate on [-oo,oo] PARI uses a different variable change that does
> not have this effect:
> 
> ? intnum(x=-oo,oo,log(x+I)/(x^2+1))
> %1 = 2.1775860903036021305006888982376139473+4.9348022005446793094172454999380755677*I
> ? Pi*log(2)+I*Pi^2/2
> %2 = 2.1775860903036021305006888982376139473+4.9348022005446793094172454999380755677*I
> 
> Cheers,
> Bill.