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Re: fundamental Integrate question


Yes I use 1/x but somehow failed to appear in the post.
Thanks a lot for your response and bringing in my attention the
setting:

Unprotect[Limit];
Limit[a___] := Null /; (Print[InputForm[limit[a]]]; False)

Best Regards
Dimitris Anagnostou

Ï/Ç Daniel Lichtblau Ýãñáøå:
> dimitris wrote:
> > Consider the classical example that incorrectly gave zero in a prior
> > version of Mathematica
> > (adopted from http://library.wolfram.com/infocenter/Conferences/5832/)
> >
> > In[345]:=
> > Integrate[f[z], {z, 1 + I, -1 + I, -1 - I, 1 - I, 1 + I}]
> > Chop[N[%]]
> > Chop[NIntegrate[f[z], {z, 1 + I, -1 + I, -1 - I, 1 - I, 1 + I}]]
> >
> > Out[345]=
> > 2*I*Pi
> >
> > Out[346]=
> > 6.283185307179586*I
> >
> > Out[347]=
> > 6.2831853071795685*I
> >
> > Of course the result is correct considering the pole at origin and the
> > Residue theorem.
> >
> > Trying to understand how Mathematica applies the Newton-Leibniz
> > formula I just want to know if
> > I am right below:
> >
> > In[511]:=
> > ((F[z] /. z -> 1 - I) - F[z] /. z -> -1 - I) + ((F[z] /. z -> 1 + I) -
> > F[z] /. z -> 1 - I) +
> >   ((F[z] /. z -> -1 + I) - F[z] /. z -> 1 + I) + (Limit[F[z], z -> -1,
> > Direction -> -I] - F[z] /. z -> -1 + I) +
> >   ((F[z] /. z -> -1 - I) - Limit[F[z], z -> -1, Direction -> I])
> >
> > Out[511]=
> > 2*I*Pi
> >
> > Thanks for any response!
> >
> > Dimitris
>
> Not clear what you use for f[x]. Maybe 1/x? Anyway, quoting the author
> of that notebook:
>
> "It is important to assess whether an integration path crosses a branch
> cut of an antiderivative (so that we might split the path into
> segments)." So yes, I would imagine it does something along the lines
> you suggest.
>
> To see explicitly what limits get computed you might first do
>
> Unprotect[Limit];
> Limit[a___] := Null /; (Print[InputForm[limit[a]]]; False)
>
>
> Daniel Lichtblau
> Wolfram Research


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