5 Jan
2009
5 Jan
'09
5:48 a.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
>>
>>
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>>
>>
>
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