RE: Integrate (undocumented feature)
- To: mathgroup at smc.vnet.net
- Subject: [mg20803] RE: Integrate (undocumented feature)
- From: "Ersek, Ted R" <ErsekTR at navair.navy.mil>
- Date: Sun, 14 Nov 1999 18:13:52 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Following my suggestion that Integrate has an undocumented capability to do
integrals around a contour in the complex plane. Bill Bertram wrote noted
cases where Integrate gives the wrong answer for such problems. One such
example was:
>
>
>Integrate[1/(x - 2I), {x, -1, 1, 1 + 3I, -1 + 3I, -1}] // FullSimplify
>
>this gives the result I Pi (wrong by a factor 2)
>
-------------------
I wonder if the problem isn't so much with the way Integrate
handles the form
Integrate[f[x],{x,x0,x1,x2, ...}]
but Integrate may incorrectly do the integral along one of the line
segments. Consider the problem below (using Version 4).
In[1]:=
Integrate[1/(x - 2I), {x, -1 + 3I, 1 + 3I}] // N
Out[1]=
0. + 1.5708*I
In[2]:=
NIntegrate[1/(x - 2I), {x, -1 + 3I, 1 + 3I}]
Out[2]=
0. - 1.5708*I
I am far from an expert on this subject, but I suspect NIntegrate got it
right and Integrate got it wrong. It seems with Version 4 we seldom hear
about errors with Integrate except when the integrand explicitly involves
complex numbers, or the limits of integration are complex. In any case I
think one should always verify results from Integrate with NIntegrate.
Regards,
Ted Ersek
For Mathematica tips, tricks see
http://www.dot.net.au/~elisha/ersek/Tricks.html