Date: Jan 21, 2013 1:07 AM Author: David Bernier Subject: Re: Are elliptic functions orthogonal? 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 doubly-periodic 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