Re: computing residues

*To*: mathgroup at smc.vnet.net*Subject*: [mg54793] Re: [mg54785] computing residues*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Wed, 2 Mar 2005 01:26:40 -0500 (EST)*References*: <200503010658.BAA25262@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

On 1 Mar 2005, at 07:58, mjumbo wrote: > > Hello, > > I would like to compute residues of analytical function, but > Mathematica's built-in function Residue produces extremely strange > results. E.g. > > Residue[1/Sin[z], {z, w}] > > gives 0, I expected it produce some conditional expression, or at > least to leave this expression unevaluated, but 0 is surely incorrect. > What I am doing wrong? > > MJM > __________ > www.newmail.ru -- óçåë ñâîáîäíûõ êîììóíèêàöèé. > > Note that: Residue[f[z],{z,w}] 0 Obviously this is meaningless. On the other hand: Residue[1/Sin[z], {z, Pi}] -1 Residue[1/Sin[z], {z, 1}] 0 Residue[1/Sin[z], {z, 0}] 1 The key seems to be the following sentence form the documentation: Mathematica can usually find residues at a point only when it can evaluate power series at that point. And indeed 1/Sin[z] + O[z] SeriesData[z, 0, {1}, -1, 1, 1] 1/Sin[z] + O[z, 1] SeriesData[z, 1, {Csc[1]}, 0, 1, 1] 1/Sin[z] + O[z, Pi] SeriesData[z, Pi, {-1}, -1, 1, 1] It looks like Residue actually takes its value from SeriesData. Since f[z] + O[z, w] SeriesData[z, w, {f[w]}, 0, 1, 1] that would explain the meaningless value 0. Certainly it is not a very satisfactory approach and Residue as well as a some other functions that have not been modified since version 2 seem overdue for an update. Andrzej Kozlowski Chiba, Japan http://www.akikoz.net/andrzej/index.html http://www.mimuw.edu.pl/~akoz/

**References**:**computing residues***From:*mjumbo <mjumbo@nm.ru>