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MathGroup Archive 2005

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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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