x↑↑x = -1 sheldonison Long Time Fellow Posts: 638 Threads: 22 Joined: Oct 2008 06/03/2014, 06:09 PM (This post was last modified: 06/03/2014, 11:14 PM by sheldonison.) (06/03/2014, 12:16 PM)tommy1729 Wrote: Somewhat of a followup : We can say D(z + theta_A(z)) = z + theta_B(z). .... Here's my interpretation of that equation. $\theta_2(z) = \alpha_2(\alpha_1^{-1}(z+\theta_1(z)))-z\;\;\; \alpha_1, \theta_1$ are the $\Im(z)>0$ Abel and theta functions, $\;\;\alpha_2, \theta_2$ are the $\Im(z)<0$ functions For Tetration, involving a pair of repelling fixed points (real bases are in this category), my conjecture is that this equation is only 1-cyclic at the real axis at the analytic boundary of both theta functions, and isn't useful anywhere else. Once you cross the real axis, I don't think the theta(z) branches follow the same path as the tet(z) branches. Basically, if you make a circle around 0, and follow $\alpha(z)$, you don't get back to where you started because there is a logarithmic branch at zero, since the Abel function is an iterated logarithm, followed by renormalization. So the path you take, for $\Im(z)<0$ and $\Im(\exp(z))<0$ for the $\Im(z)>0 \; \alpha(z)$ function has to go back into the upper half of the complex plane between z and exp(z), or else the logarithmic branches become inconsistent with a 1-cyclic theta mapping, because theta(z) has a really complicated Riemann surface branch singularity for integers at the real axis. I could try to come up with a numeric example, tet(-0.5-0.1i) and tet(0.5-0.1i), and help explain this, if you think that would help you. - Sheldon tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 06/03/2014, 08:37 PM I assume the theta singularities come from the D(z). I see no other reason. A function A with singularity at 0 and a function B with singularity at 0 implies : A(B(z)) and B(A(z)) have a singularity at 0. Hence singularities do not cancel under composition with other singularities or analytic functions. Multiple complex base tetrations ?? $\alpha_1(\text{tet}_b(z)) = z + \theta_1(z)$ $\alpha_2(\text{tet}_b(z)) = z + \theta_2(z)$ Now I had the idea $\alpha_1(\text{tet}_b(z + \theta_3)) = z + \theta_3(z) + \theta_1(z + \theta_3(z))$ $\alpha_2(\text{tet}_b(z + \theta_3)) = z + \theta_3(z) + \theta_2(z + \theta_3(z))$ And that becomes : $\alpha_1(\text{tet}_b(z + \theta_3)) = z + \theta_3(z) + \theta_4(z)$ $\alpha_2(\text{tet}_b(z + \theta_3)) = z + \theta_3(z) + \theta_5(z)$ Further simplify : $\alpha_1(\text{tet}_b(z + \theta_3)) = z + \theta_6(z)$ $\alpha_2(\text{tet}_b(z + \theta_3)) = z + \theta_7(z)$ Notice $\text{tet}_b(z + \theta_3)) = \text{tet_2}_b(z)$ , in other words another tetration function. SO we do not have uniqueness ?? Or are there branch issues again ? And how weird would it be to have a proof of non-uniqueness without a proof of existance. These equations are weird man ! In fact Im having doubts about the carleman matrices approach due to issues such as singularities and possible non-uniqueness. Still thinking ... regards tommy1729 jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 06/04/2014, 12:48 AM (06/03/2014, 08:37 PM)tommy1729 Wrote: In fact Im having doubts about the carleman matrices approach due to issues such as singularities and possible non-uniqueness. When you say the carleman matrices approach, if you are referring to the "intuitive" approach that Andrew Robbins developed (previously mentioned by others like Peter Walker I think?), then have no worries. For a real base greater than 1, it has very nice convergence properties. The convergence is slow, but it is quite well-behaved. I've been working on an analysis, based on the intuitive method, as well as my accelerated version of the intuitive method. The convergence is well-behaved for each term of the taylor series, as well as for the root-mean-squared error at any radius up to the convergence radius at the fixed point(s). For real bases greater than eta, in the principal branch, the closest singularities to the real line are at the primary fixed points, which means you can safely develop the intuitive solution using any development point on the real line, including 0 or 1. For real bases greater than 1, up to eta, you can develop from any point less than the lower fixed point. Indeed, this is just the regular iteration approach. For example, for base sqrt(2), you can develop at any point on the real line less than 2. I haven't tested, but I assume that developing above the upper fixed point is equivalent to regular iteration at that fixed point. I have my doubts about developing from a point in between. I have not had time to analyze complex bases. I have my doubts that a unique solution exists, since the periods of the fixed points are different, but I'm waiting to make an analysis before I make any claims one way or the other. ~ Jay Daniel Fox sheldonison Long Time Fellow Posts: 638 Threads: 22 Joined: Oct 2008 06/04/2014, 11:43 AM (This post was last modified: 06/04/2014, 01:29 PM by sheldonison.) (06/04/2014, 12:48 AM)jaydfox Wrote: (06/03/2014, 08:37 PM)tommy1729 Wrote: In fact Im having doubts about the carleman matrices approach due to issues such as singularities and possible non-uniqueness...... I have not had time to analyze complex bases. I have my doubts that a unique solution exists, since the periods of the fixed points are different, but I'm waiting to make an analysis before I make any claims one way or the other. Hey Jay, Welcome back. The originaly Op asked about $x\uparrow \uparrow x=i$, to which I answered that x~=2.6918099719192 + 0.62660048483655i is a solution. And from there, Tommy asked me to explain how complex base tetration works for base 3+i, so I think this has been a good thread, because I got to review some of the fascinating stuff Mike and I explored. Jay, you should see if you can generate the accelerated version of the intuitive method for base 3+i. See post #7, this thread, for some good equations and the Taylor series; I added the slog series too, and post #17, this thread for the tetcomplex.gp algorithm overview, and post #11, this thread for some pretty complex plane graphs for base=3+i, pictures. As you noted, the two pseudo periods are different in the upper and lower halves of the complex plane, as the function approaches the $\alpha_1^{-1}(x)$ inverse Abel function in the upper half of the complex plane, and $\alpha_2^{-1}$ in the lower half of the complex plane. The conjecture is that if you start with Kneser's real valued tetration, and take the function which is the first derivitive of the $f(b)=\frac{d}{dx}\text{tet}_b(x=0) dx$, then f(b) is an analytic function for the first derivative in terms of the base; This would also apply to any of the other Taylor series coefficients. Emperical results using the tetcomplex.gp pari-gp complex base program strongly support the conjecture. (06/03/2014, 08:37 PM)tommy1729 Wrote: SO we do not have uniqueness ?? Or are there branch issues again ? And how weird would it be to have a proof of non-uniqueness without a proof of existance. So for complex base tetration, uniqueness would stem from the analytic continuation from Kneser's solution, which would also answer Tommy's question about uniqueness, since Henryk Trapmann has a published proof for the uniqueness of slog for Kneser's solution from the primary fixed points, but there are multiple branches in the pseudo period of tetration, and you can get these equations, but they should all be equivalent, since real Tetration is unique. - Sheldon tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 06/04/2014, 12:22 PM (06/04/2014, 11:43 AM)sheldonison Wrote: (06/03/2014, 08:37 PM)tommy1729 Wrote: SO we do not have uniqueness ?? Or are there branch issues again ? And how weird would it be to have a proof of non-uniqueness without a proof of existance. So for complex base tetration, uniqueness would stem from the analytic continuation from Kneser's solution, which would also answer Tommy's question about uniqueness, since Henryk Trapmann has a published proof for the uniqueness of slog for Kneser's solution, but there are multiple branches in the pseudo period of tetration, and you can get these equations, but they should all be equivalent, since real Tetration is unique. Real tetration is not unique since we also have a 1periodic wave. So the question of uniqueness is imho not resolved. regards tommy1729 jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 06/04/2014, 03:48 PM (06/04/2014, 11:43 AM)sheldonison Wrote: (06/04/2014, 12:48 AM)jaydfox Wrote: I have not had time to analyze complex bases. I have my doubts that a unique solution exists, since the periods of the fixed points are different, but I'm waiting to make an analysis before I make any claims one way or the other. Hey Jay, ...Jay, you should see if you can generate the accelerated version of the intuitive method for base 3+i. Okay, I played around with my library this morning, to try to get it working with complex bases. For real bases greater than eta, my library was coded to calculate the taylor series for one singularity, then take twice the real part (based on the conjugated pair). I modified it to calculate each fixed point separately and add the results. I tested with bases e and 10 to make sure I didn't break anything. I then tried base 3+i. The solutions were initially convergent (intuitive and accelerated versions). However, as early as about size 30-40 for the matrix system, the convergence became rapid divergence. I ran into the same issue several years ago, when trying to recenter the matrix system for a real base to a complex development point. The closer the development point to the real line, the longer the initial convergence would last, before diverging rapidly. I never had time to properly analyze the problem. It wasn't an issue with numerical precision (adding precision didn't affect the divergence at all). I'll play around with it when I get time, but right now I'm still working on other tetration-related projects. For example, I'm building out my intuitive/accelerated slog library and getting the bugs out, so I can eventually upload it here. It's in SAGE/python/cython code for now, but once it's stable, I'll write a pari/gp version for those who prefer that environment. I'm also analyzing convergence (with pretty pictures!) for the intuitive and accelerated intuitive methods, in quite a bit of detail, so that takes a fair amount of time as well. ~ Jay Daniel Fox jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 06/04/2014, 04:01 PM (06/04/2014, 12:22 PM)tommy1729 Wrote: (06/04/2014, 11:43 AM)sheldonison Wrote: (06/03/2014, 08:37 PM)tommy1729 Wrote: SO we do not have uniqueness ?? Or are there branch issues again ? And how weird would it be to have a proof of non-uniqueness without a proof of existance. So for complex base tetration, uniqueness would stem from the analytic continuation from Kneser's solution, which would also answer Tommy's question about uniqueness, since Henryk Trapmann has a published proof for the uniqueness of slog for Kneser's solution, but there are multiple branches in the pseudo period of tetration, and you can get these equations, but they should all be equivalent, since real Tetration is unique. Real tetration is not unique since we also have a 1periodic wave. So the question of uniqueness is imho not resolved. regards tommy1729 For a real base greater than eta, the Kneser solution is unique. Henryk established a condition somewhere, roughly equivalent to Kneser's solution, that the tetration function should be bounded as the imaginary part goes to +/- infinity. This is equivalent to solving for an slog that asymptotically goes to a logarithm at the primary fixed points (which, coincidentally, is how I accelerate convergence of the intuitive/matrix solution). So there is a simple uniqueness criterion, for real bases greater than eta. But I'm not convinced that the Kneser solution works for complex bases. I can't imagine how to construct it, because I can't help but imagine a discontinuous first derivative at the endpoints of our initial section of the real line. (For reference, I've constructed the Kneser solution explicitly for base e, so I know how to imagine it for a real base greater than eta.) ~ Jay Daniel Fox tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 06/04/2014, 09:42 PM (06/04/2014, 04:01 PM)jaydfox Wrote: (06/04/2014, 12:22 PM)tommy1729 Wrote: (06/04/2014, 11:43 AM)sheldonison Wrote: (06/03/2014, 08:37 PM)tommy1729 Wrote: SO we do not have uniqueness ?? Or are there branch issues again ? And how weird would it be to have a proof of non-uniqueness without a proof of existance. So for complex base tetration, uniqueness would stem from the analytic continuation from Kneser's solution, which would also answer Tommy's question about uniqueness, since Henryk Trapmann has a published proof for the uniqueness of slog for Kneser's solution, but there are multiple branches in the pseudo period of tetration, and you can get these equations, but they should all be equivalent, since real Tetration is unique. Real tetration is not unique since we also have a 1periodic wave. So the question of uniqueness is imho not resolved. regards tommy1729 For a real base greater than eta, the Kneser solution is unique. Henryk established a condition somewhere, roughly equivalent to Kneser's solution, that the tetration function should be bounded as the imaginary part goes to +/- infinity. This is equivalent to solving for an slog that asymptotically goes to a logarithm at the primary fixed points (which, coincidentally, is how I accelerate convergence of the intuitive/matrix solution). So there is a simple uniqueness criterion, for real bases greater than eta. But I'm not convinced that the Kneser solution works for complex bases. I can't imagine how to construct it, because I can't help but imagine a discontinuous first derivative at the endpoints of our initial section of the real line. (For reference, I've constructed the Kneser solution explicitly for base e, so I know how to imagine it for a real base greater than eta.) Im aware of the uniqueness condition. But there is a difference between uniqueness and uniqueness condition ! In fact I proved the related TPID 4 ! http://math.eretrandre.org/tetrationforu...ght=TPID+4 At least that is how I interpret your uniqueness condition , correct me If I misunderstood. Im suggesting that as for the real tetration , also the complex base tetration that agrees on both fixpoints has a 1periodic wave ( and still agrees on both fixpoints ). Im here for quite a while. I too do not believe in a kneser solution for complex bases. Not sure if taking derivatives with respect to the base IS EQUIVALENT to kneser for complex bases ? If it is equivalent , I think I can prove with the riemann mapping that the method is doomed to fail. Yet I also do not believe that taking the derivative with respect to the base or (if different) complex kneser is EQUIVALENT to the equation with 2 theta functions. SO as far as Im concerned they might be 3 distinct methods that might or might not fail. It seems , but I could be wrong , that sheldon feels all 3 are equivalent and they work. I had the idea that we need a special type of series expansions for these theta functions ; that combination of singularities , analytic for non-integer real and periodic is something that is not well described naturally by fourier series nor by Taylor series. Also these annoying singularities make it hard to use theorems from complex analysis. The question came to me wheither or not an analogue uniqueness criterion like TPID4 could apply to complex base tetration agreeing on both fixpoints. Or complex bases tout court. Notice I dropped the idea of pseudoperiodicity , I feel it overcomplicated the equations. I have not considered loosing the property of pseudoperiodicity seriously yet. In the context of complex base tetration agreeing on fixpoints , I think the singularities are problematic for andrew robbins method if we focus on integers. If we use half-integers I think we have issues with the radius of convergeance. Welcome back mr fox regards tommy1729 jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 06/04/2014, 11:38 PM (This post was last modified: 06/04/2014, 11:45 PM by jaydfox.) (06/04/2014, 09:42 PM)tommy1729 Wrote: (06/04/2014, 04:01 PM)jaydfox Wrote: (06/04/2014, 12:22 PM)tommy1729 Wrote: Real tetration is not unique since we also have a 1periodic wave. So the question of uniqueness is imho not resolved. regards tommy1729 For a real base greater than eta, the Kneser solution is unique. Henryk established a condition somewhere, roughly equivalent to Kneser's solution, that the tetration function should be bounded as the imaginary part goes to +/- infinity. This is equivalent to solving for an slog that asymptotically goes to a logarithm at the primary fixed points (which, coincidentally, is how I accelerate convergence of the intuitive/matrix solution). So there is a simple uniqueness criterion, for real bases greater than eta. Im aware of the uniqueness condition. But there is a difference between uniqueness and uniqueness condition ! In fact I proved the related TPID 4 ! http://math.eretrandre.org/tetrationforu...ght=TPID+4 At least that is how I interpret your uniqueness condition , correct me If I misunderstood. Im suggesting that as for the real tetration , also the complex base tetration that agrees on both fixpoints has a 1periodic wave ( and still agrees on both fixpoints ). Im here for quite a while. Maybe I'm just misunderstanding what you mean by uniqueness. b^x is not the only solution to the functional equation f(z+1) = b*f(z). We can apply any 1-cyclic transform to z that we wish, e.g., b^(z+sin(2*pi*z)). Would you consider b^z to be unique? If we can't agree to a unique solution for exponentiation, there's no point talking about a unique solution to tetration or any other (inverse) Abel function. I'm suggesting that for real bases greater than eta, there is a solution for tetration that is as unique as exponentiation is. ~ Jay Daniel Fox sheldonison Long Time Fellow Posts: 638 Threads: 22 Joined: Oct 2008 06/05/2014, 01:53 PM (This post was last modified: 06/05/2014, 08:11 PM by sheldonison.) (06/04/2014, 04:01 PM)jaydfox Wrote: But I'm not convinced that the Kneser solution works for complex bases. I can't imagine how to construct it, because I can't help but imagine a discontinuous first derivative at the endpoints of our initial section of the real line. (For reference, I've constructed the Kneser solution explicitly for base e, so I know how to imagine it for a real base greater than eta.)I think a quote from Tommy best sum's up my answer: (05/31/2014, 09:23 PM)tommy1729 Wrote: But then what did mike and sheldon compute !?? There are equations to show the mathematical equivalence of the theta mappings used in kneser.gp for real bases to Kneser's Riemann mapping, assuming convergence; I posted the relevant equations below. Mike had this idea that Tetration solutions involving both fixed points could be calculated and gave some examples. So in the tetcomplex.gp program, I extending the theta mappings in kneser.gp to complex bases. Obviously there is no Riemann mapping for complex bases, but empirically there are a pair of theta mappings, which lead to a complex base solution. Lacking a Riemann mapping, this is much less robust mathematically, at least until we come up with a theoretical framework. But from a practical standpoint, it works. I published a Taylor series for base 3+I accurate to 13 decimal digits in post#7. I loaded up tetcomplex.gp with "\p 173", and "init 3+I". And after 34 iterations, it damn sure acts like a complex tetration solution for base=3+I, accurate to a little over 86 decimal digits. Each iteration improved precision by roughly 8.4 binary bits, until it reached the limit of the precision for the Abel functions and their inverses. $\alpha(\text{tet}_b(z)) = z + \theta(z)\;\;\;\theta(z) = \sum_{n=0}^{\infty} a_n (\exp(2\pi i z))^n\;\;$ formal Abel function of tet(z) equals z+theta(z) For real bases, the z+theta(z) mapping is mathematically equivalent to Kneser's Riemann mapping. Kneser's Riemann mapping function would be generated from the $a_n$ coefficients of the $\theta(z)$ mapping as follows. The Riemann mapping starts with, $z + \theta(z)\;$ which is modified too $\exp(2\pi i (z + \theta(z)))$ and then we map $z \mapsto \frac{\ln(z)}{2\pi i}$ $\text{RiemannMapping}(z)= \exp(2\pi i \times( \frac{\ln (z)}{2\pi i} + \sum_{n=0}^{\infty} a_n z^n))$ $\text{RiemannMapping}(z)= z \times \exp( \sum_{n=0}^{\infty} 2\pi i a_n z^n)\;\;$ which generates Kneser's Riemann mapping from Sheldon's theta(z) coefficients - Sheldon « Next Oldest | Next Newest »