(05/24/2011, 12:12 AM)sheldonison Wrote: I'll describe what I did to generate the Taylor series for sexp(z). Its not fancy or anything -- its just what worked for me to investigate base eta, with its parabolic convergence. I use pari-gp, to generate an interpolating polynomial. Only I center the polynomial around sexp(100), where the sexp(z) function is already converging towards e, and is fairly well behaved. Then I generate 25 points on either side of sexp(100), using 67 digits of precision.

(05/23/2011, 08:42 PM)sheldonison Wrote: (...)

Code:

a0  =  0
a1  =  1.661129667441415
a2  = -1.137387400487982
a3  =  0.841151615164940
a4  = -0.657512962174043
a5  =  0.535494578310460
a6  = -0.449853109363909
a7  =  0.387026076215351
a8  = -0.339240627153272
a9  =  0.301798047541097

(05/20/2011, 06:43 PM)andydude Wrote: That is a really interesting question. First of all, that is one of the bases for which the series expansion of (\exp_b^t(x)) in terms of x is relatively simple with "nice" coefficients, but substituting x=1 (which I think you are talking about) gives a function of t which is not strictly a power series, which makes finding that power series more difficult. Anyways, I believe I have done this before, but I don't have access to my notes right, now, so let me get back to you later today...

Andrew Robbins

(05/21/2011, 10:55 PM)sheldonison Wrote: For the results I'm posting here centered at cheta(0)=2e, I iterate the exponent of that function 95 times, to make a unit circle in the complex plane centered around cheta(0), from which a taylor series can be generated. It appears to work; I've haven't posted it before. Initialized to 67 digits accuracy in pari-gp, the algorithm seems to give results with nearly 50 decimal digits of accuracy. Here is the Taylor Series. a0=2e, printed to 32 digits.

Code:

0 5.4365636569180904707205749427053
1 1.1771399745582020467487064927981
2 0.47791083712959936964236746127117
3 0.18626062152494972692276478391796
4 0.070474191198539960880465202693624
5 0.026056306225434063913977558720610
6 0.0094541495787515083484748872855356
7 0.0033764647774015865179387607261247
8 0.0011895908149927411979137386055855
9 0.00041416349743994006206357899506395
10 0.00014268359371573690572984247219736
11 0.000048694763765091835931424063371768
12 0.000016477512451260383444394568944931
13 0.0000055326597652388183384746557130853
14 0.0000018445541337171731425492507600409
15 0.00000061095142258861804599507950586002
16 0.00000020113633929013309964268387743384
17 0.000000065845717087468591004558852969906
18 0.000000021442747870947309095492187967455
19 0.0000000069485439464512255857882560746267
20 0.0000000022412832385662916992460615895339
21 0.00000000071978893862885391677345614556987
22 2.3020973206030329181894361145544 E-10
23 7.3341040297826856350206259498267 E-11
24 2.3278852998967291568233733165642 E-11
25 7.3628505815581778431734554753314 E-12
26 2.3209857992934250244177110110812 E-12
27 7.2930204918450243443324949177246 E-13
28 2.2846097982451633980559339079833 E-13
29 7.1358033041146574466639840152247 E-14
30 2.2225543653988499920641676567938 E-14
31 6.9038181736676583445161044386747 E-15
32 2.1389425595138842758935272382177 E-15
33 6.6103445086593382475520541375449 E-16
34 2.0379986212392242975901900360689 E-16
35 6.2686731343734042989649316156714 E-17
36 1.9238630883507697992098052847583 E-17
37 5.8915865956656546031878586526468 E-18
38 1.8004488782209858871936565169317 E-18
39 5.4909672937838092357031986642134 E-19
40 1.6713366863796135261320082560915 E-19
41 5.0775249729439841733736059243593 E-20
42 1.5397061803790039357741194269061 E-20
43 4.6606303423839124073481231546437 E-21
44 1.4082983164868122075336349130200 E-21
45 4.2482389646419479670428084576358 E-22
46 1.2794035465663755925461443200159 E-22
47 3.8468885494278543290882325049087 E-23
48 1.1548705835256123872218006706955 E-23
49 3.4617530598681323751620914079008 E-24
50 1.0361306324224779753428626830104 E-24
51 3.0967381932148655440014679355037 E-25
52 9.2423246088852631003347605140667 E-26
53 2.7546045204596969212253367181553 E-26
54 8.1988428443303828566587256203397 E-27
55 2.4371065951483711807719660406315 E-27
56 7.2349910531853903620035897727885 E-28
57 2.1451384654610549641820852967050 E-28
58 6.3524081182337135717494628619063 E-29
59 1.8788780939059811381889624923360 E-29
60 5.5506897778611357646729108553389 E-30
61 1.6379251020900380753695185487417 E-30
62 4.8278087421120487722463748941123 E-31
63 1.4214279502607189609788849783519 E-31
64 4.1804970061336688633898121429008 E-32
65 1.2281981008321929711060382301444 E-32
66 3.6045845774315208753223996337347 E-33
67 1.0568098942930644804288412600335 E-33
68 3.0952921594935758380591422397535 E-34

(06/02/2011, 09:37 PM)JmsNxn Wrote: I hate to be a bit of a dunce but:
\( \text{sexp}_\eta(z) = \sum_{n=0}^{\infty} a_n (z-2e)^n \) is the correct formula for the first series correct? I only ask because this is the code I'm using (and I've also tried\( \text{sexp}_\eta(z) = \sum_{n=0}^{\infty} a_n (z+2e)^n \)) and neither seem to converge anywhere?

The only convergence I do get, is \( \text{cheta}(0) = 2e \) when I remove the plus/minus 2e.

(06/02/2011, 10:27 PM)sheldonison Wrote: There is also a lower superfunction for base eta, that I usually refer to as \( \text{sexp}_\eta(z) \), since \( \text{sexp}_\eta(0)=1 \), \( \text{sexp}_\eta(-1)=0 \), and there is a singularity at \( \text{sexp}_\eta(-2) \). \( \text{sexp}_\eta(z) \) does not grow superexponentially, but converges towards e as z increases. I recently posted the Taylor series for that function, centered at z=0, here:
http://math.eretrandre.org/tetrationforum/showthread.php?mode=threaded&tid=634&pid=5817#pid5817
- Sheldon

(06/02/2011, 10:27 PM)sheldonison Wrote: Jay suggested defining cheta(0)=2e. Then \( \text{cheta}(1)=\eta^{2e}=e^2 \), and cheta(2)=e^e, which seems like a reasonable choice for how to define cheta(0).

(06/02/2011, 11:40 PM)JmsNxn Wrote: This is more what I was looking for, thanks. And another question, do you have a similar taylor series for \( \text{slog}_\eta(z) \)? That would be the inverse of the lower super function.
......
Woah! have you ever thought to consider that since \( \text{cheta}(0) = 2*e = e + e \), \( \text{cheta}(1) = e^2 = e*e \), and \( \text{cheta}(2) = \text{sexp}_e(2) = e^e \) that the cheta function maps the growth of \( e\, \{x\}\, e \), where \( \{x\} \) is a hyper operator of x order (0 is addition, 1 is multiplication etc etc); or mathematically speaking, another conjecture:
\( \text{cheta}(x)\, =\, e\, \{x\}\, e\, =\, e\,\{x+1\}\,2 \), which should at least be true over domain [0, 2]. To see if its universally true would be very difficult, though.

(06/05/2011, 06:24 PM)JmsNxn Wrote: Thank you very much for the series approximations sheldon, but sadly the humps still occur in base \( \eta \). I'm wondering now if there is a better base to work with or if it's smarter to dump the idea of logarithmic semi-operators altogether, as they seem to be a poor extension of ackerman function to domain real.

The only interesting thing I have to report is that:
just like \( \lim_{p\to -\infty} \text{cheta}(p) = e \), \( \lim_{p\to -\infty} e\, \{p\}\, e = \ln^{\alpha -p}(2 \exp^{\alpha -p}(e)) = e \)

or that \( \lim_{p\to -\infty} \text{cheta}(p) = e\, \{p\}\, e = e \) where \( \{p\} \) is a logarithmic semi operator.

So my question was, what's the radius of convergence for the cheta series you gave me, and whats the recurrence relation so that I can produce the full \( \text{cheta}(x) \) function. I just want to test some values. for ex: if \( \text{cheta}(-1) = e + \ln(2) \) then I think we have something, but if it doesn't, oh well.