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 Complex Tetration bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 02/20/2013, 12:22 AM (This post was last modified: 02/20/2013, 12:25 AM by bo198214.) As a side remark, I want to remind you that there is no such thing like $b^z$ generally for complex b. Because of the ambiguity $e^{(\log(b)+2\pi i k)z}$, $k\in\mathbb{Z}$. I rather would talk about iteration of functions $e^{az}$ for complex $a$. I also wonder whether we already have shown that $e^{az}$ has at most one attracting fixpoint, i.e. exactly one if $e^a$ is in the Shell-Thron-Region, and none outside. Balarka Sen Junior Fellow Posts: 25 Threads: 7 Joined: Feb 2013 02/20/2013, 07:14 AM mike3 Wrote:Yes, I forgot about that. Robbins' method does not in fact work for every base. I am not really concerned for the base. In my case, it is enough if it works for all the positive integer base. I am much more interested in complex (or perhaps real) heights. (A. Robbins is here too? Mein gott, this seems like a whole community of mathematicians!) mike3 Wrote:tetration is not defined for negative-integer heights equal to -2 or less. It has branch points (not poles) at those heights. Thanks. But why the code gives such results? Is it doing something like regularization for negative integers smaller than -1? What if we define those point by taking principle values of those (if possible)? mike3 Wrote:I don't know of a code off-hand, but I suppose I could whip one up. I terribly need some code like that since I don't have mathematica or maple. Please create one if possible. mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 02/20/2013, 07:59 AM (This post was last modified: 02/20/2013, 08:03 AM by mike3.) (02/20/2013, 07:14 AM)Balarka Sen Wrote: mike3 Wrote:Yes, I forgot about that. Robbins' method does not in fact work for every base. I am not really concerned for the base. In my case, it is enough if it works for all the positive integer base. I am much more interested in complex (or perhaps real) heights. (A. Robbins is here too? Mein gott, this seems like a whole community of mathematicians!) Then you'll probably like the Kneser method more than Robbins' one. Robbins' only works for a limited range of complex heights since it creates a Taylor series in the height, which is limited by the singularities (and also, quasi-limited by the area of very rapid growth). (02/20/2013, 07:14 AM)Balarka Sen Wrote: mike3 Wrote:tetration is not defined for negative-integer heights equal to -2 or less. It has branch points (not poles) at those heights. Thanks. But why the code gives such results? Is it doing something like regularization for negative integers smaller than -1? What if we define those point by taking principle values of those (if possible)? What commands are you using to give tetration? What commands did you use to get that value? As when I use it, PARI/GP returns a "*** log: zero argument in mplog" error -- which is what I'd expect, since there's a singularity. I use "init(2)" then "loop()" then "sexp(-3)". Perhaps you missed a command or used the wrong one? (02/20/2013, 07:14 AM)Balarka Sen Wrote: mike3 Wrote:I don't know of a code off-hand, but I suppose I could whip one up. I terribly need some code like that since I don't have mathematica or maple. Please create one if possible. I'll see if I can write one. I'm curious: what exactly are you trying to do with this, anyway? Balarka Sen Junior Fellow Posts: 25 Threads: 7 Joined: Feb 2013 02/20/2013, 11:30 AM mike3 Wrote:Then you'll probably like the Kneser method more than Robbins' one. Robbins' only works for a limited range of complex heights since it creates a Taylor series in the height, which is limited by the singularities (and also, quasi-limited by the area of very rapid growth). But the problem with kneser is that it requires some time to calculate the polynomials and set-up with the base. I need a code that can continuously output sexp(b, z) for different bases. For example, if I use code Sheldon created and want to calculate sexp(2, 0.5) + sexp(3, 0.5) then it isn't possible to directly command PARI to do that. We need to write init(2);sexp(0.5) and init(3);sexp(0.5) differently and then sum the results up. But consider that we need to do this several times. Then it is impossible by this algorithm. But I think Robbins' method doesn't requires the use of initializing the base(?). It's okay if limited, real values would do the thing. mike3 Wrote:What commands are you using to give tetration? What commands did you use to get that value? As when I use it, PARI/GP returns a "*** log: zero argument in mplog" error -- which is what I'd expect, since there's a singularity. I use "init(2)" then "loop()" then "sexp(-3)". Perhaps you missed a command or used the wrong one? I am using the code "kneser.gp" that I downloaded from here somewhere. It doesn't gives me any error, just returns the complex value I posted before. I typed init(2) then loop(6) and at last sexp(-3). It returns the same, ~6.73 + i*4.53 mike3 Wrote:I'm curious: what exactly are you trying to do with this, anyway? I mentioned before that I am working on a function. But I am currently too shy to give it away since I haven't succeeded to get any possible properties of the function sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 02/20/2013, 12:04 PM (This post was last modified: 02/20/2013, 12:16 PM by sheldonison.) (02/20/2013, 11:30 AM)Balarka Sen Wrote: ... I am using the code "kneser.gp" that I downloaded from here somewhere. It doesn't gives me any error, just returns the complex value I posted before. I typed init(2) then loop(6) and at last sexp(-3). It returns the same, ~6.73 + i*4.53Its just a round off error. For approximations near the real axis, the code is using a Taylor series approximation for sexp(0). Then when you type in sexp(-3), it figures out that it should use sexp(0), and then take the logarithm three times. You initialized for base(2). But, the iterative method is not an exact solution, so sexp(0) is not exactly 1, but more like 1+4.524E-63. This is just an approximation artifact of the $\log_2(\log_2(\delta))$, where $\delta$=6.527E-63. The real part is just the $\log_2(206.6)$. The imaginary part of 4.532i is just $\frac{\pi i}{\log(2)}$. As a side note, its interesting to calculate values for sexp in the range from -3 to -2, and note that they all have this same imaginary value. And if you graph the approximation, you can clearly see that it has a singularity as z approaches -3, with the slope getting very large. I agree with Mike, that especially for results in the complex plane, the kneser.gp code is your best bet. For larger values of imag(z), >0.85, it switches to a complex theta mapping generated from the complex fixed point schroder function, since the Taylor series at the real axis is only accurate to 32 decimal digits within a unit circle around the origin. See my recent post at http://math.stackexchange.com/questions/...-tetration for a really good short concise description, which happens to be for base(2), which you are using. Good luck with your explorations! - Sheldon Balarka Sen Junior Fellow Posts: 25 Threads: 7 Joined: Feb 2013 02/21/2013, 01:52 PM (This post was last modified: 02/21/2013, 01:53 PM by Balarka Sen.) Funny thing is that if we use Robbin's method, then the time it takes to compute for integer bases is longer than it takes to compute non-integer bases! sheldonsion Wrote:especially for results in the complex plane, the kneser.gp code is your best bet So, can we somehow suppress the output kneser.gp produces when we enter init(base)? sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 02/21/2013, 02:23 PM (This post was last modified: 02/21/2013, 05:24 PM by sheldonison.) (02/21/2013, 01:52 PM)Balarka Sen Wrote: So, can we somehow suppress the output kneser.gp produces when we enter init(base)? Here it is! I kneserquiet.gp   kneserquiet.gp (Size: 34.6 KB / Downloads: 489) It has a global variable, quietmode, which I defaulted to '1' in this version. I also defaulted the precision to "\p 28", instead of "\p 67". This makes the program very fast. init(2) takes about 0.4 seconds on my computer. Using "\p 28" is good for about 14 decimal digits precision. If you want, you can also type in "\p 67", prior to typing in init(base), which is good for about 32 decimal digits precision, but takes about 5 seconds for init(2). - Sheldon Code:\r kneserquiet.gp ... help menu prints out ... gp > init(2); gp > sexp(0.5) %3 = 1.4587818160364 « Next Oldest | Next Newest »

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