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fundamental Integrate question

• To: mathgroup at smc.vnet.net
• Subject: [mg73031] fundamental Integrate question
• From: "dimitris" <dimmechan at yahoo.com>
• Date: Tue, 30 Jan 2007 06:44:16 -0500 (EST)

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

• Follow-Ups:
  • Re: fundamental Integrate question
    • From: Murray Eisenberg <murray@math.umass.edu>
  • Re: fundamental Integrate question
    • From: Daniel Lichtblau <danl@wolfram.com>