Superroots (formal powerseries)  Printable Version + Tetration Forum (https://math.eretrandre.org/tetrationforum) + Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) + Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) + Thread: Superroots (formal powerseries) (/showthread.php?tid=375) Pages:
1
2

Superroots (formal powerseries)  Gottfried  10/25/2009 Recently I found Andrew's remark in "designing a tetration library", that the superrots were not yet well developed. Facts on superroots seem to be spread over various threads; so to have some collection under an expressive title I've put together some details, tending to compile more information from time to time as they appear. (@admin: maybe that msg is better located in some related thread, for instance the tetration library thread) Ok, I've put the text in plain text and am lazy to MimeTex it today, perhaps I'll rework it next days. ====================================================================== A short collection concerning superroots ====================================================================== Starting point is the nice powerseries for Code: ´ g(x) = (1+x)^(1+x)  1 Using the exponential/logarithmseries for this we write first Code: ´ g(x) = exp( log(1+x)*(1+x))  1 and get Code: ´ g(x) = 1*x + 1*x^2 + 1/2*x^3 + 1/3*x^4 + 1/12*x^5 + 3/40*x^6 This series has the nice property, that the constant term vanishes and also, that the linear term has coefficient 1, so g(0)=0 and g'(0)=1, so we can do some common operations with it: inversion, iteration, ... getting exact coefficients. Now we define higher orders by something like chaining, which is not exactly iteration of g(). The sequence of functions Code: ´ g(x,1) = (1+x)1 gives similarly nice shaped powerseries, for instance Code: ´ g(x,3) = (1+x)^(1+x)^(1+x) 1 From this it is easy to define a sequence of functions for exponentialtowers of integer heights: Code: ´ f(x,h) = g(x1,h) + 1 = x^x^x^...^x // hoccurences of x Note, that this is in principle all well known and is merely a restatement of known results. The unusual aspect with that sequence of powerseries is, that the leading coefficients stabilize when the height increases, and thus we have a "strange" behave when the height increases to infinity. Example: we get the following table of coefficients (where the rows contain the coefficients for one height and each column is associated with one power of x): Code: ´ 0 1 0 0 0 0 0 0 0 ... where the first column (containing zeros) represent the placeholders for the nonexistent constant terms. (The first row was appended to get a meaningfully interpretation for the "once"iterate; it represents just g(x,1) = (1+x) 1 . The limit case for h>inf begins with the same coefficients as the last row of the table above)  Inversion Since the g(x,h)series have no constant term but a linear term with unitcoefficent, we can invert each of that gseries. Expressed by the appropriate ffunction we get the superrootpowerseries for each integer height. Let's denote the inverse functions as gi() and fi(), then for gi(x,2) we get Code: ´ gi(x,2) = x  x^2 + 3/2*x^3  17/6*x^4 + 37/6*x^5  1759/120*x^6 and a higher h, for instance Code: ´ gi(x,12) = x  x^2 + 1/2*x^3 + 1/6*x^4  3/4*x^5 + 131/120*x^6 which again stabilizes for h>inf Code: Table of coefficients for gi(x,h), h=1.. ====================================================================== The h'th superroot ====================================================================== The shown powerseries, formally seen, give the functons for the h'th superroots: Code: ´ fi(x^x,2) = x = gi(x^x1,2)+1 and the computation of the h'th superroot can be implemented by calls of the gi(x,h)function: Code: ´ ssrt(x,h) = fi(x,h) = gi(x1,h) + 1 This gives, if convergent, the base b, which must be exponentiated h times to equal the given value x.  Convergence: Concerning the range of convergence I don't have an idea yet. For instance for g(x,2) we can guess a rate of decrease similar to µ/k^2 where k is the index and µ some constant, so we should have a range of convergence for x<=1 only. For f(x,2) consequently we had then 0<x<=2 . It seems, that the occuring divergences are not "too strong" so that we can extend the domain for x using Eulersummation to get meaningful approximations even if only 64 or 128 terms are known.  Interpolation to fractional orders: The special form of the powerseries, where each k'th coefficient becomes constant when chainingheight h>=k this poses a new challenge for the interpolation to fractional heights. I have currently no idea how to proceed here...  (should be continued) Gottfried RE: Superroots (formal powerseries)  andydude  10/26/2009 Nice job. Might I also link to this thread, where I posted a document with some superroot stuff. RE: Superroots (formal powerseries)  Gottfried  10/26/2009 (10/26/2009, 01:50 AM)andydude Wrote: Nice job. Might I also link to this thread, where I posted a document with some superroot stuff. Thanks! Yes, my idea were to just collect the links to relevant threads/posts here to have a starting place for readers who want to step into the problem of the superroot. I'll just start a list of links; would be good to be extended. Gottfried Superroot: List of relevant msg's/threads  Gottfried  10/26/2009 A.Robbins On Analytic iteration... (2009) See pages 1821; some more formulae are derived there "limit of selfsuperroots is exp(exp(1))..." (msg 200910) initial conjecture (msg 200910) proof a curiosity/paradox(?) with limit of consecutive orders of superroots (msg 200902) superroots of real numbers x>e Connection between superroot and LambertW: (msg 200803) Lambert W function and the Super Square Root (msg 200803) Deriving tetration from selfroot? I.Galidakis Hyperroot with Lambert W (2009) This discussion gives also code for maple for some investigations Ideas for superroots of higher order 2019 Aug: MSEquestion on superroots 2019 Aug: My own answer for generalization of the LambertWformula for the higher order superroots A small treatize on the generalization of LambertW on my webspace (2015/2017) (referred to in my MSEanswer) RE: Superroots (formal powerseries)  robo37  10/28/2009 Dudes can someone put some of this information on Wikipedia? On this page? You see people seem to want that article to be merged with the one about tetration and as I can't think of any more information about the function I seem to be fighting a loosing argument. RE: Superroots (formal powerseries)  Gottfried  10/28/2009 (10/28/2009, 09:20 AM)robo37 Wrote: Dudes can someone put some of this information on Wikipedia? On this page? You see people seem to want that article to be merged with the one about tetration and as I can't think of any more information about the function I seem to be fighting a loosing argument. But this is so far only "original research" while the encyclopedic approach of wikipedia focuses on citations of standard material (refereed journals, books which have undergone a full criticism and publishing process). As far as we cannot locate such material concerning "superroot" independently of the broader context of tetration we cannot help in this case. I myself am unfortunately especially useless for this since I have little access to relevant literature (resp time to acquire relevant articles & to understand the professional lingo) and only recall only marginally that/where I've read here&there about such x^x and superrootapplications in praxi (as I stated in wikipedia the text about "wexzal" and the superroot for the calculation of explosiondriven projectile and automobilemoving). So, well, having said this I'll better hand your request over to the next one... Gottfried Superroots article "WexZal"  Gottfried  10/29/2009 I've uploaded that "WexZal"article; it is originally in plain text (around 1998, if I recall right) but I thought it might be better to have it in .pdf. In a private mail the author welcomed to distribute this text. I'll add that statement later (I'll put it into Henryk's lib, too) [attachment=616] A treatize of iterated exponentiation and inverse, seemingly initially inspired by the question of superroot of order 2. Mentions two practical applications RE: Superroots article "WexZal"  bo198214  10/29/2009 (10/29/2009, 05:56 AM)Gottfried Wrote: I've uploaded that "WexZal"article; it is originally in plain text (around 1998, if I recall right) but I thought it might be better to have it in .pdf. In a private mail the author welcomed to distribute this text. I'll add that statement later (I'll put it into Henryk's lib, too) Thanks a lot Gottfried! Why is this not published as a proper book? I think there should be included some more data, like finishing date and some jacket text about the authors. And great: it contains appliations, as this really often asked and some ignorant physicists just think that there can not be applications. RE: Superroots article "WexZal"  Gottfried  10/29/2009 (10/29/2009, 12:08 PM)bo198214 Wrote: Why is this not published as a proper book?Well I know nearly nothing around that text; if I google today for the authors I only find a reference at Rob. Munafo's site without presenting further reference  that's all. Here is a msg of si.math, which I saved when I came across it: see also google groups Code: Betreff: Online "book" about solution of y=x^x Code: Betreff:Re: Wexzal book RE: Superroots article "WexZal"  bo198214  10/29/2009 Thank you Gottfried for the additional information. However in the book the relation between Lambert W and superroot/coupled root is not described, which was mentioned by James Knight here. 