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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/
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