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Re: Are elliptic functions orthogonal?
Posted:
Jan 21, 2013 1:07 AM


On 01/20/2013 10:04 PM, Sycho wrote: > This just in to the alt.2600 news room. On Sun, 20 Jan 2013 17:22:01 > 0800 (PST) it was announced to all in a public briefing, Vaughan > Anderson<vaughan.andursen@gmail.com> made the following declaration > and shocked the world when the following was announced: > >> On Jan 18, 10:28 pm, h...@work.pk (Sycho) wrote: >>> This just in to the alt.2600 news room. On Fri, 18 Jan 2013 14:58:51 >>> 0800 (PST) it was announced to all in a public briefing, Jeremy >>> Sample<vaughan.andur...@gmail.com> made the following declaration and >>> shocked the world when the following was announced: >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>>> Can an arbitrary function be uniquely expanded in a series solution of >>>> elliptic integrals? >>> >>>> That is to say, can you apply an algorithm like the Fourier analysis, >>>> (or Bessel, Legendre, etc.) to an arbitrary function, using elliptic >>>> integrals instead of trigonometrics as the basis function? >>> >>>> I wonder if this could be a useful technique for reducing nonlinear >>>> data, in systems where certain, simple cases are known to have >>>> elliptic solutions. >>> >>>> Your scholarly input would be greatly appreciated, even if it means >>>> referring me to journal articles, as long as they're by specific >>>> authors. >>> >>>> TIA. >>> >>> They can be whatever you want them to be so long as you pay them >>> enough "hush" money. >>> >>> Cookies also help. >> >> What is it with you and cookies? The holidays are over, and it's time >> to start on your New Year's resolution to lose weight. ;) > > The dark side *always* has cookies. It's an unwritten law. > >> I believe that elliptic integrals (or functions), in general, are not >> orthogonal, and therefore not suitable for use in series expansions. >> But since they are not a discretely indexed function like Trigs, >> Bessels, Legendres, etc. it may be possible to *find* values for the >> elliptic control parameter that make them orthogonal, which would >> impose a discrete index onto the functions. > > Everybody's got to believe in something. I, OTOH believe I'll have > another beer. [..]
The Weierstrass P(w1, w2) is a doublyperiodic function with periods w1 and w2 in the complex plane:
"Weierstrass's elliptic function"
Wikipedia: "The Weierstrass elliptic function can be given as the inverse of an elliptic integral."
Also, P(w1, w2) seems to me to have a pole of order two at the origin.
I'd bet on Vaughan Anderson's story ...
dave



