Re: Why does Limit sometimes just return what I input?

• To: mathgroup at smc.vnet.net
• Subject: [mg44605] Re: Why does Limit sometimes just return what I input?
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Mon, 17 Nov 2003 03:38:40 -0500 (EST)
• Organization: The University of Western Australia
• References: <bp28u9\$351\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <bp28u9\$351\$1 at smc.vnet.net>, jmonk <jmonk at hep.man.ac.uk>
wrote:

> I am trying to find the limit of some gamma matrix traces when a small
> parameter \[Epsilon]->0.  I am using the package Tracer.m

Note that I picked up an error/omission in this package when I loaded it
in. In the SortLine routine there is the following:

If[ list === rlist, ,
If[ Sort[list] === Sort[rlist],

Now, since the (extended) syntax of If is

If[condition, t, f, u] gives u if condition evaluates to neither True
nor False.

then the meaning associated with the repeated comma is rather important.

I note that the pakcage at

http://library.wolfram.com/infocenter/MathSource/2987/

has a revision date of 1991.

> and I define 4 lines:
>
> Spur[l1, l2];
> line11 = G[l1, p1, p3, G5, n, {Î½}, (p1 + k), {Î¼}];
> line12 = G[l1, p1, p3, G5, n, {Î¼}, (p3 - k), {Î½}];
> line21 = G[l2, p2, p4, G5, np, {Î»}, (p2 - k), {Î¼}];
> line22 = G[l2, p2, p4, G5, np, {Î¼}, (p4 + k), {Î»}];

Your message came through with some very strange characters. Perhaps you
could re-post in a readable format?

> The vectors n and np are well defined by
>
> OnShell[on,{n, xx_,\[Chi]1. xx + Î»\_1**\[Chi2p.xx]},
> 	{np, xx_, lim[Î¶1 . xx + Î»\_\[Zeta]2p.¹*¶xx}];
>
> with \[Chi]1, \[Chi]2p, Î¶1and\[Zeta]2p similarly defined. The momenta
> p1...p4 also have their dot products defined. n and np depend on a small
> parameter \[Epsilon], and it turns out that the highest order terms in the
> lines line11...line22 are of order 1/\[Epsilon].
>
> I want to take the limit e.g.
>
> Limit[line11*\[Epsilon],\[Epsilon]->0,Direction->-1]

Instead of taking the limit (using Limit), it is usually preferable to
perform a series expansion using Series.

> and then contract the 2 lines on the \[Mu] Lorentz index, but usually when
> I try to take the limit it just returns the limit unevaluated.
> Sometimes, however, it does return the limit.  I can use the function
> Residue to get the answer, but I want to know why Limit does not work.

Well, Residue (or Series) may be more appropriate anyway. In my
experience, Limit is not "as solid" as Series.

> The expressions for the lines are quite long - there are 338 terms, so I
> wonder if it is too long for mathematica to cope with unless all of the
> computer resources are available to it.  I am running Mathematica 5.0 on
> Mac Os X, G4 1Ghz 256 Mb.
>
> The full script I have is too long to include, but I could forward it if
> requested.

Perhaps you could post the relevant input expressions or give a URL to
where the Notebook can be obtained.

Cheers,
Paul

--
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au
AUSTRALIA                            http://physics.uwa.edu.au/~paul

```

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