|WikiProject Mathematics||(Rated C-class, High-importance)|
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)
I moved the following from the subject page:
- 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,
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 18.104.22.168 (talk) 14:19, 8 September 2010 (UTC)
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: