Re: fundamental Integrate question

*To*: mathgroup at smc.vnet.net*Subject*: [mg73048] Re: [mg73031] fundamental Integrate question*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Wed, 31 Jan 2007 00:04:41 -0500 (EST)*References*: <200701301144.GAA14308@smc.vnet.net>

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

**References**:**fundamental Integrate question***From:*"dimitris" <dimmechan@yahoo.com>