Talk:Weierstrass's elliptic functions

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The homogenity relation doesn't seem to work for all c since cτ may not remain in the upper half plane. I don't quite understand how this relation can serve to define doubly periodic functions with arbitrary period pairs.

It would probably be better to choose parameter Λ = periodic lattice initially, rather than τ. Taking the reciprocal of τ would fix up the imaginary part, and that's taking the basis for periods in the other order. That in fact is just a special case of the PSL (2,Z) action that is implicit in choice of general basis, and which means τ can be taken to be in the usual fundamental domain within the upper half plane. But I'd agree this is tough on the reader - does two steps at once. Charles Matthews 19:31, 10 Feb 2004 (UTC)

Also, the statement that all doubly periodic functions with given periods form a field C[P, P'] is not quite clear to me. Does it mean that any such function can be expressed as a rational function in P and P'? AxelBoldt 23:57, 9 Feb 2004 (UTC)

Given a period lattice Λ, what is true is that all meromorphic functions periodic under Λ form a field that is actually C(P, P'), i.e. rational functions in the Weierstrass P and its derivative. The ring notation C[P, P'] would stand for polynomials in P and its derivative; this is enough to generate the functions with singularities only at the points of Λ. To get from there to the general result requires an argument, I guess, like this:

- the general function F will have a finite number of poles, mod Λ

- construct directly a function Gw having a simple pole at 0 and general point w, only;

- given F, subtract off some translates of P and derivative to get a function only with simple poles, and then express that as a linear combination of functions Gw, plus a function with no singularities;

- an elliptic function with no singularity is constant by Liouville's theorem.

The usual construction of a Gw function is as a difference of Weierstrass zeta-functions. This looks like the most serious step.

Charles Matthews 08:34, 10 Feb 2004 (UTC)

I've made some minimal changes to sort out the page. In a sense I think this page should be about Pe-related formulae, and the general elliptic function and elliptic curve theory should live somewhere else.

Charles Matthews 08:57, 10 Feb 2004 (UTC)

The Weierstrass P function should have a double pole at z=0, right? your definition seems to be missing that term. should it read

\mathcal{P}(z;\tau) =\frac{1}{z^2} + \sum_{n^2+m^2 \ne 0}{1 \over (z-n-m\tau)^2} - {1 \over (n+m\tau)^2}

This correction and some of the other comments are good, but it seems to me in many ways the article has been messed up and dumbed down from the way I left it. Why drag in the old-fashioned half-period formulation, and then say things inconsistent with it? What was wrong with sticking to the modular formulation? A lot of stuff has been replaced with a regurgitation of old-fashioned points of view and formulas which do not related to the article on theta functions. Gene Ward Smith 07:36, 20 July 2005 (UTC)

Hi, Welcome back to Wikipedia (as I notice you've been gone a long time). I've been editing the article on and off; its possible that some of what you are complaining about are edits I've made, for which I'm sorry. As to "old-fashioned points of view", I'm not sufficiently advanced on the topic to be able to tell the difference between old and new. I know I've worked "from the literature"; if I tripped across a relation that seemed useful, I'd add it to the article. I admit that this is not a very elegantly structured article, which has bugged me, but I haven't had any vision to make any overhauls, with the exception of modular discriminant, which I wanted to make into its own stand-alone article. linas 14:34, 20 July 2005 (UTC)
Oh, and as to "dumbed down", be aware that there has been ooodles of screaming about "too complicated" in many other math articles. A policy I've tried to hew to, is that, for an article like this, an average college student with a good grounding in calculus (or an average professor with absolutely no prior experience at all with elliptic/modular functions) should be able to read at least half-way through the article before getting mired. Thus, Charles Matthews definitions regarding a field C[P,P'] that is a ring of polynomials over C is good, but should occur somewhere other than the beginning of the article, as it would hopelessly throw the beginners for a loop. The current article seems readable; if its lacking a more modern or more formal or a more high-brow definition, it should be added (but I don't know what that might be). Hope I'm not sounding too bombastic here. linas 14:51, 20 July 2005 (UTC)


Hi. Good points made above by Linas. Well, I gotta respond to dumbed down, an allegation that I refute utterly. An article is dumbed down iff it misses important conceptual points so as to be palatable to the lowest common denominator, or perhaps makes mistakes in the interests of simplicity.

The article is not dumbed down by these criteria.

However, I'd be happy to concede that extra material (quite possibly not understandable by Linas's professor on the Clapham omnibus) might be added. I'd say that Gene Ward Smith would be particularly well placed to add such material.

Best wishes Robinh 20:13, 20 July 2005 (UTC)

Contents

[edit] minus sign

I believe the integral expression for P(z) is off by a minus sign. Bateman and Mathematica both have the integral going from +Infinity to P(z), i.e. backwards.

[edit] Symbol history?

Was Weierstrass the first to use his P? What is the source of the symbol? Is it some form of a German version of P or is it something else?



[edit] Periods or half-periods?

In the section about the ei, the ωi change their meaning from periods to half-periods without much warning, it is confusing.--Bernard 22:27, 6 August 2006 (UTC)

Hi Bernard. You are absolutely right. I was just about to start changing the article to be self-consistent, but then thought it couldn't hurt to check whether people would prefer the ωi to be the periods or the half periods. My vote would be for the ωi to be half periods. What do other editors reckon? Robinh 20:16, 7 August 2006 (UTC)

My vote is for periods, it seems more natural to me, but I have no textbook to compare with. --Bernard 21:07, 7 August 2006 (UTC)
I would be very happy to have the omegas as periods everywhere, which is the convention I was brought up on (and my doctorate was in this area). Charles Matthews 21:37, 7 August 2006 (UTC)

[edit] Pe-function?? Peh-function??

can someone please explain what a pe-function is supposed to be? I've seen this thing called p-function zillions of times, but never pe-function.--345Kai 03:22, 5 April 2007 (UTC)