Talk:Laurent series

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I wonder if it is useful to allow for infinitely many negative-exponent terms, so that functions with essential singularities can be described? You can't talk formally about those of course. AxelBoldt, Thursday, July 11, 2002

The problem is that if you allow for infinitely many negative-exponent terms, Laurent series stop being unique. For instance:

1 + 1/x + 1/x2 + 1/x3 + ... = 1/(1 - 1/x) = -x/(1 - x) = -x(1 + x + x2 + x3 + ...) = -x - x2 - x3 - x4 - ...

— Josh Grosse, Thursday, July 11, 2002

But the left hand series converges for |x| > 1, while the right hand side converges for |x| < 1, so the two Laurent series don't really overlap. AxelBoldt

One can certainly discuss such series, but I've never seen them referred to as "Laurent series". Of course, if you have infinitely many terms of negative degree but only finitely many terms of positive degree, (and if c = 0), then you have a "Laurent series in 1/z". But as far as I can tell, this is no more special than a Laurent series in any other expression. — Toby Bartels, Friday, July 12, 2002

In algebra, formal Laurent series are certainly restricted to finitely many negative indices, since that's the only thing that makes sense. In complex analysis however, things seem to be different:

I guess it's kind of nice if you can define the residual of a function simply by pointing to the -1 coefficient of its Laurent series; otherwise you need a case distinction for poles vs. essential singularities. AxelBoldt

Upon further consideration (also involving looking in some textbooks), you're absolutely right, Axel. — Toby Bartels, Friday, July 12, 2002

The article talks about formal Laurent series and appears to allow infinitely many negative degree terms. I don't think it is possible to define those in any meaningful way, if one wants a ring at least. AxelBoldt 14:46 Aug 13, 2002 (PDT)

Hey, you're the one that wanted these terms in the first place! Of course, I realise that you didn't mean that in the formal case. There may be a way to cure this since we need to identify certain infinitely negative series with certain infinitely positive ones (as per Josh Grosse's example above). I'll have to think about it. Meanwhile, I've asked a colleague that apparently knows about this how it works. — Toby 13:47 Aug 14, 2002 (PDT)

The book Complex Variables and Applications by Ruel V. Churchill of the University of Michigan allows an infinite number of negative terms, and I see no reason why the definition should not allow for an infinite number of negative terms, as this would allow for expansions of √x and lnx around the singularities at 0 that converge for all x, as well as a infinitely convergent series for |x| around the singularity. Speaking of which, if anyone does know about these, have they ever heard of the ones mentioned above?Scythe33 01:16, 10 July 2005 (UTC)

[edit] Merge with z transform

Should this article be merged with Z-transform? --Richard Clegg 14:39, 24 August 2006 (UTC)

No I think. The Z-transform is a very important topic in signal processing, and while it does use Laurent series, the primary focus of that article is not the series in itself, rather, how a function changes under the Z transform and how this can be applied to solving problems. In short, the Z-transform is much more than just an application of Laurent series. Oleg Alexandrov (talk) 16:08, 24 August 2006 (UTC)