Talk:Weierstrass elliptic function

Early article discussion

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 G_w 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 G_w, plus a function with no singularities;

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

The usual construction of a G_w 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}\neq 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)[reply]

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)[reply]

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)[reply]

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)[reply]

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. — Preceding unsigned comment added by 129.105.28.16 (talk) 18:43, October 21, 2005‎

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?

Periods or half-periods?

In the section about the $e_{i}$ , the $\omega _{i}$ change their meaning from periods to half-periods without much warning, it is confusing.--Bernard 22:27, 6 August 2006 (UTC)[reply]

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 $\omega _{i}$ to be the periods or the half periods. My vote would be for the $\omega _{i}$ to be half periods. What do other editors reckon? Robinh 20:16, 7 August 2006 (UTC)[reply]

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)[reply]

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)[reply]

More than 5 years have passed and this issue has not been fixed!

The overwhelming consensus in the literature is that the periods of the lattice should be $2\omega _{i}$ . This is a convention that dates back centuries. See Whittaker-Watson, or DLMF.

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)[reply]

What about me, the end user

As my daughter used to say. Not to be "down and dumb" but "what's it for" ? ( cool graphic on unit disk) Does this have applications (used to represent xxxx )in electricity, chemistry, structural analysis, natural phenomenon. DG12 (talk) 20:31, 10 November 2011 (UTC)[reply]

in fact, elliptic functions have plenty of applications (in particular, to electricity, starting from the classics), but this is not the place to discuss them (see WP:Talk page guidelines). Sasha (talk) 01:48, 11 November 2011 (UTC)[reply]

Well I guess I will never know:"1 day for US$39.00" DG12 (talk) —Preceding undated comment added 01:19, 14 November 2011 (UTC).[reply]

An inconsistancy.

The section on the modular discriminant $\Delta$ is not consistant as currently written. The section defines the discriminant in terms of the $g_{2}$ and $g_{3}$ coefficients, which are in turn defined in terms of general periods. Thus we have $\Delta (\omega _{1},\omega _{2})$ . The discriminant section, however, claims that $\Delta =(2\pi )^{12}\eta ^{24}$ . This is not correct. It is $\Delta (1,\tau )$ that is equal to $(2\pi )^{12}\eta ^{24}$ . The power $\eta ^{24}$ is a modular invariant. The discriminant is a modular form of weight 12.

I don't want to meddle with the writing in case I make things worse. A regular page maintainer should do it.

Mike Stone (talk) 14:23, 19 April 2012 (UTC)[reply]

Two conventions for modular discriminant

The definition of modular discriminant as given now is incostistent with the convention used at Modular form#The modular discriminant and Ramanujan tau function. I think the convention used in those two pages (without the $(2\pi )^{12}$ factor) is more common, however both of them appear in literature, and the convention with the factor used here arguably makes more sense in the context of elliptic functions. However this inconstistency should be resolved somehow, and mention of the existence of two different conventions in literature should be added (perhaps both here and at Modular form#The modular discriminant). But I don't wanna make this decision unilaterally and would appreciate some feedback from regular maintainers of this page. GreenKeeper17 (talk) 18:53, 6 April 2017 (UTC)[reply]

Weierstrass Constant

What's about the Weierstrass Constant 0.474949...? 46.115.82.168 (talk) 14:40, 31 May 2013 (UTC)[reply]

Missing parameter in section "The constants $e_{1},e_{2}$ and $e_{3}$ "

In the section "Invariants", expressions for the two independent quantities $g_{2}$ and $g_{3}$ are given in terms of $\tau$ and $\omega _{1}$ . However, in the section "The constants $e_{1},e_{2}$ and $e_{3}$ ", the dependence on $\omega _{1}$ is dropped. There should be two independent parameters describing $e_{1},e_{2}$ and $e_{3}$ (because $e_{1}+e_{2}+e_{3}=0$ ), but there is only $\tau$ . I can see two possible resolutions. Either explicitly state that $\omega _{1}=1$ and $\omega _{2}=\tau$ , or even better, include a second parameter (such as $\omega _{1}$ ) so that all cases are covered. Yzarc314 (talk) 14:06, 25 July 2017 (UTC)[reply]

I second that. Also, in the same section the formulas for $e_{2}(\tau )$ and $e_{3}(\tau )$ in terms of $a$ , $b$ , $c$ should be interchanged if one wants to have the ordering $e_{1}>e_{2}>e_{3}$ mentioned below when the Weierstrass roots are real. This can be easily checked for instance numerically. Ludius (talk) 21:09, 5 January 2018 (UTC)[reply]

Past discussion about the symbol ℘

A pointer: there was discussion about the symbol ℘ in Talk:Weierstrass p#Cleanup, now a redirect to this page. -- Teika kazura (talk) 00:06, 18 September 2018 (UTC)[reply]

General Theory typo

" if c is any non-zero complex number,

{\displaystyle \wp (cz;c\tau )=\wp (z;\tau )/c^{2}} "

This is surely wrong, it is inconsistent with a similar rule where in the second argument the whole lattice is multiplied by c (not just the second basis element).Createangelos (talk) 11:28, 27 October 2018 (UTC)[reply]

Fixed now.Createangelos (talk) 07:57, 20 November 2018 (UTC)[reply]

Suggestion: Split paragraph into Short paragraph and short list

Instead of listing three concepts within one paragraph, split them out into a list. This lets the physical structure of the text better match the conceptual structure.

Currently: The Weierstrass elliptic function can be defined in three closely related ways, ...<snip>..., for fixed z the Weierstrass functions become modular functions of τ.

I suggest: The Weierstrass elliptic function can be defined in three closely related ways, each of which possesses certain advantages:

   1) As a function of a complex variable z and a lattice Λ in the complex plane. 
   2) In terms of z and two complex numbers ω1 and ω2 defining a pair of generators, or periods, for the lattice.
   3) In terms of z and a modulus τ in the upper half-plane. This is related to the previous definition by τ = ω2/ω1, which by the conventional choice on the pair of periods is in the upper half-plane. Using this approach, for fixed z, the Weierstrass functions become modular functions of τ.

Michael McGinnis (talk) 02:24, 31 January 2020 (UTC)[reply]

Suggestion: If you are too tired or low-energy or pressed for time ...

then please let someone else edit Wikipedia.

Case in point:

" Since the square of the derivative of Weierstrass' elliptic function equals the above cubic polynomial of the function's value, $\wp '\left({\frac {\omega _{i}}{2}}\right)^{2}=\wp '\left({\frac {\omega _{i}}{2}}\right)=0$ for $i=1,2,3$ ."

The "above cubic"? The "above cubic"?

I looked all over and there were so many third powers that it is ridiculous to expect the reader to be able to figure out what the author was thinking.

If you're going to refer to an equation or anything else that a) is nowhere near your reference text and b) is not labeled at all ... then please: say what you mean.

Thanks.2600:1700:E1C0:F340:4CA2:1162:525:8896 (talk) 23:47, 25 September 2019 (UTC)[reply]

Please explain symbols

In the Definition section, the author has committed the ubiquitous but ever annoying sin of introducing a quantity (here, $\lambda$ ) which is not... defined. Writing Wikipedia articles like this ensures that no-one who really needs the article (i.e. who actually needs to learn about the subject) has any use of it. Please explain what $\lambda$ is if you are using it. Wdanbae (talk) 09:20, 18 January 2022 (UTC)[reply]

@Wdanbae: In

\wp (z,\Lambda ):={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \smallsetminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right),

\lambda

is just an index, denoting individual elements of the lattice

\Lambda \setminus \{0\}

. Similarly, in

-\ln(1-z)=\sum _{n\in \mathbb {N} }{\frac {z^{n}}{n}},

n

is just an index, denoting individual elements of the natural numbers. A1E6 (talk) 13:56, 18 January 2022 (UTC)[reply]

Wrong expressions for g_2 and g_3

The expresions for g_2 and g_3 in this article state that they have a q-expansion that has only even powers, which is not true. 128.41.61.181 (talk) 17:23, 9 June 2023 (UTC)[reply]

Addition theorem of the Weierstrass's integral of the third kind

$\int _{0}^{u+v}{\frac {\mathrm {d} t}{\wp \left(t\right)-\wp \left(a\right)}}=\int _{0}^{u}{\frac {\mathrm {d} t}{\wp \left(t\right)-\wp \left(a\right)}}+\int _{0}^{v}{\frac {\mathrm {d} t}{\wp \left(t\right)-\wp \left(a\right)}}+{\frac {1}{\wp '(a)}}\ln \left({\frac {\begin{vmatrix}1&\wp (u)&\wp '(u)\\1&\wp (v)&\wp '(v)\\1&\wp (a)&-\wp '(a)\end{vmatrix}}{\begin{vmatrix}1&\wp (u)&\wp '(u)\\1&\wp (v)&\wp '(v)\\1&\wp (a)&\wp '(a)\end{vmatrix}}}\right)$
2409:8A3C:4322:8670:88E8:63:3694:89E0 (talk) 16:06, 19 June 2024 (UTC)[reply]

Definition

@Rocco Riccii I noticed you changed my revision back again because I supposedly "didn't take the homogeneity of the function into consideration".

The homogeneity, i.e., https://mathworld.wolfram.com/WeierstrassEllipticFunction.html Eq. (78), is not that relevant in defining $\wp (z,\tau )$ as the "shorthand notation" of $\wp (z,1,\tau )$ . At best it might be helpful to show why $\wp (z,1,\tau )$ and $\wp (z,\omega _{1},\omega _{2})$ are related. But even then it would be: $\wp (z,\omega _{1},\omega _{2})={\frac {1}{\omega _{1}^{2}}}\wp ({\frac {z}{\omega _{1}}},1,\tau )$ . Hence, your edit does not use the correct scaling factor.

In any case it would be better to state the homogeneity property under the subsection "Properties" rather than to apply it without introducing it first.

Also, when you disagree with a revision of your edit, please start a disccusion on the talk page first before reverting it back again to prevent WP:EDITWAR.

Kind regards, Roffaduft (talk) 07:37, 13 March 2025 (UTC)[reply]

You literally wrote the same thing i wrote in the scaling factor just the other way around, i wrote

\wp (z,\tau )=\omega _{1}^{2}\wp (\omega _{1}z,1,\tau )

while you wrote

\wp (z,\omega _{1},\omega _{2})=\omega _{1}^{-2}\wp ({\frac {z}{\omega _{1}}},1,\tau )

which are literally the same thing. And notation wise I believe it's better to show the actual relation between two different notations as many people learn on this platform and don't have access to other books or sites that clarify things like this. So a lack of correct relations can lead to ambiguity. Rocco Riccii (talk) 09:37, 13 March 2025 (UTC)[reply]

It is "literally" not the same thing, but regardless..

What you wrote down is a particular example of the property: https://mathworld.wolfram.com/WeierstrassEllipticFunction.html Eq. (78), without the property being introduced anywhere. It is therefore premature; seemingly appearing out of thin air.

Ergo: add it to the "Properties" subsection, first the general statement and then the specific example as mentionned above. Otherwise it's just not informative.

Kind regards, Roffaduft (talk) 11:03, 13 March 2025 (UTC)[reply]

You tried to correct something and wrote down the same thing so yes you literally write the same thing. Additionally the link you keep adding from wolfram alpha does not mention at all the case involving the period ratio. It does not "appear out of thin air" it simply follows from the properties of the the P function like its homogeny. Rocco Riccii (talk) 11:22, 13 March 2025 (UTC)[reply]

It does not "appear out of thin air" it simply follows from the properties of the the P function like its homogeny

Oh? And where on the Weierstrass elliptic function wikipedia page can I find said property? That's the point. You've made a change without any context or reference. Regardless if it's correct, that is just not the way to make edits. With no added context, it violates WP:NOR IMHO.

Also,

\wp (z,\tau ):=\wp (z,1,\tau )

contained the "is defined as" symbol, not the equality symbol. That is, it just gives a definition to the shorthand notation

\wp (z,\tau )

.

So, again, just add the general property to the correct subsection and maybe give

\wp (z,\omega _{1},\omega _{2})=\omega _{1}^{-2}\wp ({\frac {z}{\omega _{1}}},1,\tau )

(or which ever version you prefer) as an example.

Kind regards, Roffaduft (talk) 11:38, 13 March 2025 (UTC)[reply]

Well many things aren't directly mentioned in Wikipedia articles and still formulas appear. It is simply deducible by the lattice expansion of P and additionally the article mentions the homogeneity of g2 and g3. I'm sorry to see that it bothers you to write something that improves clarity for new readers. Either way, best regards and hope you understand the point of my edit. Rocco Riccii (talk) 11:55, 13 March 2025 (UTC)[reply]

There is no consensus. You haven't explained why your edit doesn't violate one of the core principles of wikipedia, i.e., WP:NOR.

@Tito Omburo, would you be so kind to provide a third opinion here?

Kind regards, Roffaduft (talk) 12:04, 13 March 2025 (UTC)[reply]

It does not violate any principle pf Wikipedia. Please stop this nonsense. It is easily derivable from the lattice expression of P so there isn't much to discuss about. I repeat that the purpose of this edit is to clarify a definition that can lead to confusion and ambiguity among new readers. Have a good day. Rocco Riccii (talk) 12:18, 13 March 2025 (UTC)[reply]

$\wp (z,\tau )=\wp (z,1,\tau )$ is the correct definition of $\wp (z,\tau )$ . It seems like the correct formula you want to write is actually $\wp (z,\omega _{1},\omega _{2})=\omega _{1}^{-2}\wp (z/\omega _{1},\omega _{2}/\omega _{1})$ . Tito Omburo (talk) 12:28, 13 March 2025 (UTC)[reply]

Which is the same as I wrote but you simply brought the ω1^2 on the other side as substituted z with z/ω1. In that section the article defines what ℘(z,τ) is. So what is written is simply highlighting ℘(z,τ) instead of ℘(z,ω1,ω2). What you are writing is the same thing that i am stating:

Writing ℘(z,ω1,ω2)=ω1^(-2)℘(z/ω1,1,ω2/ω1) is the same as what I wrote ℘(z,1,ω2/ω1)= ω1^2℘(ω1z,ω1,ω2). So I don't really get what your point is Rocco Riccii (talk) 12:43, 13 March 2025 (UTC)[reply]

The main issue is not whether your application of the "homogeneity property" is correct, it is about the way you introduce it. You ignore the "is defined as" symbol, you don't introduce the property first, you don't provide a reference, etc.

You just state the equation, then claim that it is obvious and therefore shouldn't be reverted. That's just textbook WP:NOR. The fact that we have to have a discussion about the correct application of the "homogeneity property" in the first place means it's clearly not WP:CALC. I'd suggest adding it to the "Properties" subsection, but I'm fine with @Tito Omburo solution as well.

Given that you only made 15 edits so far, I assume you're new to wikipedia. It might be beneficial to read up on WP:NOR, WP:VER and WP:NPOV when you make your next edit in order to WP:AVOIDEDITWAR.

Kind regards, Roffaduft (talk) 12:54, 13 March 2025 (UTC)[reply]

I've read the both guidelines and appreciate your attention. Although if the formula can be derived doing 2 steps of calculations from a formula that is already presented in the article (in this case the lattice expansion) it seems pretty obvious that there is no need of citation etc. Either way I think that the revision that Tito Omburo did sets a perfect boundary on this discussion. Have a good day. Rocco Riccii (talk) 13:18, 13 March 2025 (UTC)[reply]