Talk:Elliptic function
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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)
Vandalism
The page has been vandalised.
Charles Matthews 06:03, 9 Sep 2003 (EDT)
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)