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 Does anyone have taylor series approximations for tetration and slog base e^(1/e)? bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/24/2011, 10:38 AM (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. You mean, you take the interpolation polynomial at the integer-points of sexp? Yeah true, that also yields regular iteration. I think we discussed this a lot with Ansus - the Newton (interpolation) formula. Gottfried Ultimate Fellow Posts: 776 Threads: 121 Joined: Aug 2007 05/24/2011, 02:22 PM (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  =  0.301798047541097Well, now I'm surprised. I've get that coefficients -however accurate only up to 6 digits, but maybe they converge if I use higher precision - by the most simple eigen-decomposition of the 32x32 carleman matrix. While the formal h'th powers of the carleman-matrices occur if I raise the diagonalmatrix of the eigenvalues to the h'th power, Pari/GP is able to convert this to a powerseries in x, if I enter the indeterminate x instead of a explicite h-value for the powers. (Note that this is the fourth method in my short treatize on "four methods of interpolation") Well, Sheldon's method seem to allow much more precision and I do not see yet, how I could reproduce this by simply increasing the Pari/GP-resources in decimal precision and matrix-size. Gottfried Gottfried Helms, Kassel JmsNxn Long Time Fellow Posts: 380 Threads: 78 Joined: Dec 2010 05/26/2011, 03:18 AM (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 Your close to what I'm talking about. I'll write it out in full math format $ f(t) = a\, \{t\}\, b = \left\{ \begin{array}{c l} \exp_\eta^{\alpha t}(\exp_\eta^{\alpha-t}(a) + \exp_\eta^{\alpha -t}(b)) & t \in [-1,1]\\ \exp_\eta^{\alpha t-1}(b * \exp_\eta^{\alpha 1-t}(a)) & t \in [1,2]\\ \end{array} \right.$ which maps the growth of addition to multiplication to exponentiation. I hope that base eta will behave better than base 2 and base e behaved. JmsNxn Long Time Fellow Posts: 380 Threads: 78 Joined: Dec 2010 06/02/2011, 09:37 PM (This post was last modified: 06/02/2011, 09:48 PM by JmsNxn.) (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 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. sheldonison Long Time Fellow Posts: 663 Threads: 23 Joined: Oct 2008 06/02/2011, 10:27 PM (This post was last modified: 06/02/2011, 11:53 PM by sheldonison.) (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.Hey James, Hopefully, this helps. The correct formula is $\text{sexp}_\eta(z) = \sum_{n=0}^{\infty} a_n (z^n)$ Cheta(z) is the upper entire superfunction of eta, which grows superexponentially as z increases. Unfortunately, how to define cheta(0) is a somewhat arbitrary, since cheta(z) is always bigger than e, as z goes to minus infinity. 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). Centered around 0, the series I posted will generate these values. More recently, Henryk and Dimitrii have written a paper where the upper $\text{SuperFunction}_\eta(0)=3$. Perhaps there will eventually be a reason to pick a definitive value, but that hasn't happened yet. 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/tetrationforu...17#pid5817 - Sheldon JmsNxn Long Time Fellow Posts: 380 Threads: 78 Joined: Dec 2010 06/02/2011, 11:40 PM (This post was last modified: 06/02/2011, 11:41 PM by JmsNxn.) (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/tetrationforu...17#pid5817 - Sheldon 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. (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). 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. sheldonison Long Time Fellow Posts: 663 Threads: 23 Joined: Oct 2008 06/03/2011, 12:48 AM (This post was last modified: 06/03/2011, 12:51 AM by sheldonison.) (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.It is an interesting sequence. Here is the series, centered at z=1, $\text{slog}_\eta(z) = \sum_{n=0}^{\infty} a_n (z-1)^n$ Code:a0=   0 a1=   1.6364055628757310098612069643305 a2=   1.0153219515675015927934054348231 a3=   0.60179106451341218323473861841285 a4=   0.35339094138233716197130631267800 a5=   0.20678022805222642138569218148044 a6=   0.12077316515278617589978715489636 a7=   0.070466597739279935817347004921549 a8=   0.041088245566444697493192432669165 a9=   0.023947858691724371628412673016534 a10=  0.013953603992552690728627252149490 a11=  0.0081285329698961693398261557991908 a12=  0.0047344290157003070913140831631485 a13=  0.0027572046775250114750438564059555 a14=  0.0016055653508202650722548361996376 a15=  0.00093487396331912588479860322878510 a16=  0.00054431539381581329680504421391946 a17=  0.00031690239123736519247306265808875 a18=  0.00018449370377075905076738832103314 a19=  0.00010740430864046301276907787649912 a20=  0.000062524232227043156013995698181196 a21=  0.000036396822638059218270554447632005 a22=  0.000021186957652934110380738251130135 a23=  0.000012332895471487133422245016435244 a24=  0.0000071788340691456493380397479368256 a25=  0.0000041786510411693011327284124119868 a26=  0.0000024322734042405353969581364435823 a27=  0.0000014157397670132180924517455634643 a28=  0.00000082404283673548797945624589827040 a29=  0.00000047963619402863627513438889010921 a30=  0.00000027917101829902645542548058586310 a31=  0.00000016248949999299272203056483378299 a32=  0.000000094575181457179592966349170760430 a33=  0.000000055046061997819873417493220126620 a34=  0.000000032038543589826489735755462890716 a35=  0.000000018647341967554525035509582154381 a36=  0.000000010853228813698946831281796207325 a37=  0.0000000063168274147666049763200649142710 a38=  0.0000000036765226692784746720709040758475 a39=  0.0000000021398030836117500316624962487179 a40=  0.0000000012453998812027015780706541464576 a41=  0.0000000007248404052267904098654234587018 a42=  0.0000000004218661070727302103010309894460 a43=  0.0000000002455306059973809636590738454709 a44=  1.4290106986271220339032492794899 E-10 a45=  8.3169529782673189838917026421766 E-11 a46=  4.8405198310832728584491149379096 E-11 a47=  2.8172073806752434116210055497734 E-11 a48=  1.6396258116384352659166197756211 E-11 a49=  9.5426686705080144354640068061238 E-12 a50=  5.5538502152136607519127775538049 E-12 a51=  3.2323452797056846816888732034200 E-12 a52=  1.8812245466465630762639581584087 E-12 a53=  1.0948708034538274092484245688059 E-12 a54=  6.3721280701164076567624351207206 E-13 a55=  3.7085619145590230253906096801015 E-13 a56=  2.1583706178992941955046700446313 E-13 a57=  1.2561629614172847097262503849877 E-13 a58=  7.3108092357697884172557590560006 E-14 a59=  4.2548519124264617256971973371109 E-14 a60=  2.4762985306030106868904605580111 E-14 a61=  1.4411896206100053414406968180449 E-14 a62=  8.3876219415829221560909589871987 E-15 a63=  4.8815324822646088477541099811656 E-15 a64=  2.8410125546371166957177683205131 E-15 a65=  1.6534450727661439551155006062390 E-15 a66=  9.6229018805487868709553017018049 E-16 a67=  5.6004383866369015723533210751934 E-16 a68=  3.2594001769485484340769051661933 E-16 a69=  1.8969376400216642523206535563610 E-16 a70=  1.1039976379052598207387127395496 E-16 a71=  6.4251454988114292160593966066487 E-17 a72=  3.7393621961593764263505452461775 E-17 a73=  2.1762653817172712533134127189169 E-17 a74=  1.2665604966549427015232899034031 E-17 a75=  7.3712271908237475668169947042325 E-18 JmsNxn Long Time Fellow Posts: 380 Threads: 78 Joined: Dec 2010 06/05/2011, 06:24 PM (This post was last modified: 06/05/2011, 08:02 PM by JmsNxn.) 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. sheldonison Long Time Fellow Posts: 663 Threads: 23 Joined: Oct 2008 06/05/2011, 08:47 PM (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.Cheta(-1)=e*(log(2)+1). cheta(z-1)=$\log_\eta(\text{cheta}(z))=e*\log(\text{cheta}(z))$ cheta(z+1)=$\exp_\eta(\text{cheta}(z))=\exp(\text{cheta}(z)/e)$ Cheta(z) is entire, but the series convergence for a finite number of terms is limited by how close we are to the region of superexponential growth, and how many terms are used. It probably has an effective radius of convergence of about 2 with the number of terms I posted. If you want more convergence as real(z) increases or decreases, use the recurrence relation below. If you want more convergence as imag(z) increases, one way to get that is to generate the series for cheta(z-100), centered at -100. For cheta/sexpeta, I actually use Newton Polynomial interpolation, centered around -100 for cheta, and +100 for sexpeta, with a 50 term polynomial, which has pretty good convergence out to a radius of 25 (~28 digits), and is accurate to 50 digits within a unit radius. I also recently included cheta/sexpeta support in my latest kneser.gp program. - Sheldon « Next Oldest | Next Newest »

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