Re: Problème avec intnum

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

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.