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