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MathGroup Archive 2007

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Re: fundamental Integrate question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73053] Re: [mg73031] fundamental Integrate question
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Wed, 31 Jan 2007 00:24:45 -0500 (EST)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <200701301144.GAA14308@smc.vnet.net>
  • Reply-to: murray at math.umass.edu

What's f[z]???

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
> 

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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