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