6 Jan
2009
6 Jan
'09
10:37 p.m.
>> Interesting!
>>
>> I encountered the LerchPhi function recently, too. Take the standard Gregory
>> series for Pi/4 = 1 - 1/3 + 1/5 ... and introduce powers of Sinc into each term:
>>
>> Define
>> f[k_, x_] := Sum[ Sinc[(2n-1)x]^k * (-1)^(n-1)/(2n-1), {n, 1, Infinity}]
>>
>> Then f[0, x] = Pi/4 for all x.
>>
>> MMA 7 expresses f[1, x], f[2, x] etc., in terms of the Lerch function.
>
> Mma tends to Lerch munstrously when polylog would suffice,
> and polylogs have more relations. For f[1,x], Macsyma and I get
>
> inf
> ==== n
> \ (- 1) sin((2 n - 1) x)
> > -----------------------
> / 2
> ==== (2 n - 1) 2 %i x 2 - %i x
> n = 1 log (- %i %e ) - log (- %i %e )
> - ----------------------------- = ---------------------------------------
> x 8 x
>
> and you may have better luck with trilogs for f[2], etc. (Caution:
> 7.0 has a d/dk(Li[k](...)) numerics bug.)
>
> Speaking of psychoanalytic, the moment Mma Lerched, I inadvertently
> recalled the Addams counterpart while trying to retrieve "Lurch".
> Somehow error-correcting this has absolutely erased the Addams
> name from my brain. All that remain are trespassers Grimace and
> Hamburglar.
>
> --rwg
> Why, oh, why didn't they name their daughters Desicca?
>
>> I can prove that f[1, x] = Sum[ Sinc[(2n-1)x] * (-1)^(n-1)/(2n-1) ] equals Pi/4
>> for x in [-Pi/2 , Pi/2]. This means that, for those x, we can multiply each
>> term of the Gregory series by Sinc[(2n-1)x] without changing the sum.
>
>> I conjecture that for k = 1, 2, 3, ..., f[k, x] equals Pi/4 for x in [-Pi/(2k) ,
>> Pi/(2k)].
>>
>> (Was Lerch in the Addams family, or was it the Munsters?)
>>
>> Bob Baillie
>> --------------------
>>
>> rwg@sdf.lonestar.org wrote:
>>> Mma 7.0 just startled me by turning the Fourier series for the line
>>>
>>> Pi
>>> -- (4 Pi - 3 t + I Sqrt[3] t), 0 <= t <= 2 Pi,
>>> 3
>>> into
>>> -I t 2
>>> LerchPhi[E , 2, -]
>>> 3 (I t)/3 I t 1
>>> L(t):= --------------------- + E LerchPhi[E , 2, -].
>>> (2 I t)/3 3
>>> E
>>>
>>> But L(t + 2 Pi) = E^(2 Pi/3) L(t). I.e., translating by 2 Pi
>>> *rotates* by 120 degrees! Eh? Sure enough, plotting L(t),
>>> 0 < t < 6 Pi, draws a perfect equilateral triangle. There seems
>>> to be such a relation among n-1 Lerchs for each regular n-gon.
>>> Some simple consequence of n-secting the series? Psychoanalytic
>>> continuation.
>>> --rwg
>>> PS, Veit Elser's difference map algorithm,
>>> http://en.wikipedia.org/wiki/Difference_map_algorithm , has become only
>>> the second entity to solve the 82% Arnold Dozenegger disk packing puzzle
>>> completely unaided. (Not counting Emma Cohen, who got massive clues from
>>> Emma Cohen.) Also, it's clear that Rod Stephenson's clustering algorithm
>>> will do it, probably in ~1 hr --way longer than Veit's, who clearly has
>>> something dangerous.
>>> -------
>>> Merriam-Webster's Unabridged:
>>> Main Entry: prince albert
>>> Usage: usually capitalized P&A
>>> Etymology: after Prince Albert Edward (later Edward VII king of England)
>>> [...] 2 : a man's house slipper with a low counter and goring on each side
>>>
>>> ALGORISMIC MICROGLIAS
>>>
PS (With the help of my laptop overheating) I forgot to add that the 7.0
Sinc doc exhibits, in effect, this strangissimo sequence:
In[6]:= Table[2/Pi*Integrate[Product[Sinc[x/k], {k, 1, 2*n - 1, 2}],
{x, 0, Infinity}], {n, 8}]
467807924713440738696537864469
Out[6]= {1, 1, 1, 1, 1, 1, 1, ------------------------------}
467807924720320453655260875000
(unattributed, but A068214.)
This may be simply an artifact of excessive haste. Specifically,
3636.98 mph. (Actually, the last term is
7
491
1 - --------------------.)
3 12 6 7 6 6
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