       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}, 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/

```

• Prev by Date: Re: MathLink
• Next by Date: Re: Mathematical Experiments (how to construct more functions)
• Previous by thread: computing residues
• Next by thread: Re: computing residues