Re: Integrate (undocumented feature)
- To: mathgroup at smc.vnet.net
- Subject: [mg20787] Re: Integrate (undocumented feature)
- From: "Bill Bertram" <wkb at ansto.gov.au>
- Date: Thu, 11 Nov 1999 00:22:55 -0500
- Organization: Australian Nuclear Science and Technology Organisation
- References: <80asap$it2@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Ersek, Ted R wrote in message <80asap$it2 at smc.vnet.net>...
>Earlier I wrote about the following results using Version 4.
>-------------------------------------
>
>
>The documentation for NIntegrate says:
>"NIntegrate[f, {x,x0,x1, ... ,xk}] tests for singularities at each of the
>intermediate points xi. If there are no singularities, the result is
>equivalent to an integral from x0 to xk. You can use complex numbers xi to
>specify an integration contour in the complex plane."
>
>Although the documentation doesn't say so it seems this applies to
Integrate
>as well. When I wrote the previous email I was thinking of the line above,
>but didn't remember that I read this in the documentation for NIntegrate
not
>Integrate.
>
>Below I give convincing evidence that this works with Integrate. Here I
>integrate along a closed contour in the complex plane. Notice I get the
>same answer when I apply a theorem related to Residues.
>
>In[1]:=
>Integrate[1/(z^2+4),
> {z,-1,1,1+3I,-1+3I,-1}]
>Out[1]=
>Pi/2
>
>
>In[2]:=
>(2 Pi I)Residue[1/(z^2+4),{z,2I}]
>Out[2]=
>Pi/2
>I think the documentation for Integrate should be changed to mention this
>feature.
Or perhaps it should be avoided altogether!
Consider the following,
Integrate[1/(x - 2I), {x, -1, 1, 1 + 3I, -1 + 3I, -1}] // FullSimplify
this gives the result I Pi (wrong by a factor 2)
And upon changing the contour (still closed and around the singularity)
Integrate[
1/(x - 2I), {x, -1 - I, 1 - I, 1 + 3I, -1 + 3I, -1 - I}] // FullSimplify
the result is 0.
Not the sort of results to inspire confidence in Mathematica's integration
methods!
(NIntegrate does give the correct result for both contours however)
Cheers,
Bill