Re: f( x + O(x) ) or f(x) + O(x) ?

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

On Sat, Feb 20, 2021 at 03:36:37PM +0000, Jacques Gélinas wrote:
> In a previous post, I used this oneliner to get the vector of 
> Maclaurin coefficients of a function:
> 
> maclv(f,k) = Vec( f( x+O(x^(k+1)) ) );
> 
> The boundary case k=0 should return f(0) as in
> maclv( cosh, 0) == [1]
> maclv( sinc,  0) == [1]
> 
> But this yields an incorrect result for this modified Riemann Xi function,
> 
> Xi(x) = my(s=1/2+x); Pi^(-s/2)*gamma(1+s/2)*(s-1)*zeta(s);

zeta is wrong:

? zeta(1/2+x+O(x))
%32 = -1.9884831127532564396510949587415908792*x+O(x^2)
? zeta(1/2+x+O(x^2))
%33 = -1.4603545088095868128894991525152980125-3.9226461392091517274715314467145995137*x+O(x^2)

The real bug is

? 1 + O(x) == O(x)
%1 = 1

The first "bad" commit is this one:

commit 55a851fb843be85f15b46231a1431f2ffaef84da
Author: Karim Belabas <Karim.Belabas@math.u-bordeaux.fr>
Date:   Thu Mar 12 19:07:24 2020 +0100

    65- problems with gequal(t_SER or t_POL) [#2188]

    inexact coefs or const wrt. some variables.
    E.g. S=x+O(x^2);T=S+O(y); S == T -> 0

Cheers,
Bill