Talk:Elliptic function

Lattice

We need an article on fundamental pair of periods that reviews all of the properties of a 2D lattice so that this article and the modular forms article (and the Jacobi & Wierestrass elliptic articles) can reference it. linas 05:10, 13 Feb 2005 (UTC)

See my comment at modular form. Charles Matthews 08:17, 13 Feb 2005 (UTC)

Weierstrass

I moved the following from the subject page:

An elliptic function on the complex numbers is a function of the form

E(z; a,b) = ∑_m∑_n (z - m'a -n'b)^-2

where a and b are complex parameters and m and n range over the integers. As written, this series is improper and divergent; but it can be made convergent by taking the Cauchy principal value, which is the limit as x->∞ of the sum of those terms with |z - m'a - n'b| < x.

The function is periodic with two periods, a and b. Plotting E(z) on x versus E'(z) on y results in an elliptic curve.

A real elliptic function can also be defined in the same way. Either a is real and b imaginary (in which case the elliptic curve has two parts, E(z + b/2) being also real for real z) or a + b is real and a - b is imaginary (in which case the elliptic curve has one part).

Degenerate elliptic functions and curves are obtained by setting a or b to infinity. If a or b is infinite, but not both, the Cauchy principal value diverges and other means must be used to define the function. If both are infinite, E(z) is simply 1/z². If a is real and b is infinite, the curve consists of one smooth part and one point. If a is imaginary and b is infinite, the curve is a loop that crosses itself. If both are infinite, the curve is the semicubical parabola x³ = y²/64.

The formula for E is wrong I believe, and there are certainly other elliptic functions. I don't know how to rescue this. AxelBoldt 01:48 Nov 8, 2002 (UTC)

I just picked up the yellow book. The correct formula is

E(z; a,b) = z^-2 + ∑_m∑_n (z - m'a -n'b)^-2-(n'b)^-2,

where n=m=0 is excluded from the sum. I think it should be put at Weierstrass's elliptic function. -phma

References

The elliptic functions as they should be in the references is eccentric. Better for example to go to Whittaker & Watson, though their notation is not what the modern standard is (same for all the older books). Tannery and Molk is the classic reference; book by Weber. But the old books are out of print, I suppose - more's the pity. Charles Matthews 22:14, 19 Nov 2004 (UTC)

Definition and Properties

Layman question. Should a' = p a + q b and b' = r a + q b instead read as a' = p a + q b and b' = r a + s b? It seems odd to calculate s and then throw it out. It also seems to leave a degree of freedom, which allows for arbitrary a' and b'. (unsigned anonymous user, 15 August 2005)

Yes, that is correct, it was a typo in the formula. linas 21:14, 15 August 2005 (UTC)

Different layman question: why is multiplication denoted by a space, instead of using the multiplication symbol or the middle dot (× or · respectively, both listed as common in the wiki article on multiplication)? a' = p·a + q·b in complex analysis context (as opposed to algebraic context) is semantically clearer and unambiguous. -- Pomax, 8 September 2010 —Preceding unsigned comment added by 130.161.177.89 (talk) 14:19, 8 September 2010 (UTC)

Historical note

The article states: "Historically, elliptic functions were first discovered by Carl Gustav Jacobi..." Well, whoever wrote this should definitely read the article "Niels Henrik Abel" by G.Mittag-Leffler( who sure knew what he was talking about!), in which it is proved beyond the shadow of a doubt that the real originator of the theory of elliptic functions is Abel and not Jacobi. Mittag-Leffler's text is available(in French) at the following URL: Niels Henrik Abel Gemb47 (talk) 12:32, 12 November 2012 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Elliptic function/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Really fundamental in the development of analysis, but the article does not yet reflect this. Only just start in my view, but feel free to adjust the rating and replace this comment. Geometry guy 21:39, 20 May 2007 (UTC)

Last edited at 21:39, 20 May 2007 (UTC). Substituted at 02:02, 5 May 2016 (UTC)

Simplifying lead and definition: doubly periodic and meromorphic

For more experienced editors: it seems to me that the description and definition of elliptic functions can be simplified and made more explicit by simply saying that they are (defined as) doubly periodic meromorphic functions. In particular, rather than "elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions", I suggest "elliptic functions are doubly periodic meromorphic functions", or "elliptic functions are meromorphic functions which are also doubly periodic", etc. In the definition, it can say something like "Elliptic functions are complex functions which are doubly periodic and meromorphic. That is, <insert definitions of doubly periodic and meromorphic>", or "A complex function $f\colon \mathbb {C} \to \mathbb {C}$ is an elliptic function if it is doubly periodic and meromorphic", etc.

But I'm not sure how appropriate this is, hence this talk topic. Kclisp (talk) 22:53, 25 March 2023 (UTC)