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tetration from alternative fixed point - sheldonison - 05/23/2010
Has anyone generated the base e tetration from an alternative fixed point? The second fixed point (as far as I can tell) for base e is 2.0622777296+i*7.5886311785 It seems like one could develop a complex plane super function from this alternative fixed point. And from the complex plane super function, one could use one of the methods (Riemann mapping or the Cauchy method) to turn this complex super function into a real valued function with f(0)=1, and with singularities at f(-2), f(-3), etc. This would be a different tetration solution, but it would be analytic. Just curious if anyone had tried this. - Shel RE: tetration from alternative fixed point - bo198214 - 05/23/2010
(05/23/2010, 12:05 AM)sheldonison Wrote: Has anyone generated the base e tetration from an alternative fixed point? The second fixed point (as far as I can tell) for base e is 2.0622777296+i*7.5886311785 The problem with alternative fixed points is that we dont have a sickel between them. I.e. if we connect two conjugated non-primary fixed points with a straight line, then the image under \( e^z \) of this straight line overlaps itself. This is due to the imaginary part of the fixed point pair is apart more then \( 2\pi \) which makes the image of any connecting line between these fixed points, revolve around 0 at least once. For Kneser's solution we need a region that is bounded by a line connecting the two fixed points and the image of this line. Perhaps one (you?) could prove, that regardless how you connect a conjugated fixed point pair, that is not the primary one, the image of this line intersects itself or the connecting line; i.e. both lines never delimit a singly connected region. RE: tetration from alternative fixed point - sheldonison - 05/24/2010
(05/23/2010, 07:54 AM)bo198214 Wrote: The problem with alternative fixed points is that we dont have a sickel between them. I.e. if we connect two conjugated non-primary fixed points with a straight line, then the image under \( e^z \) of this straight line overlaps itself. This is due to the imaginary part of the fixed point pair is apart more then \( 2\pi \) which makes the image of any connecting line between these fixed points, revolve around 0 at least once.Would this refer to the graph of the complex super-function, generated from the secondary fixed point, or to the graph of the real valued super-exponential, after the Kneser solution? Quote:For Kneser's solution we need a region that is bounded by a line connecting the two fixed points and the image of this line. I'd like to graph the 3*pi*i contour line of the super-function generated from the secondary fixed point, 2.0622777296+i*7.5886311785 The super-function "grows" away from the fixed point, and the first n*pi*i contour encountered is the 3*pi*i contour. This is analogous to the primary fixed point, which eventually reaches the pi*i contour. The theory is that the contour line would have real values from -infinity to +infinity, and that the exponent of that contour line, would trace out the real values from -infinity to zero, and the next exponent would trace out the real values from zero to one etc. It is straightforward to generate the super-function from the secondary fixed point, but the inverse super-function is giving me difficulties, and I need to get the iteration equations for the inverse super-function working before I can graph the 3*pi*i contour, and then perhaps I will understand why this contour line does not allow for Knesser's construction. RE: tetration from alternative fixed point - bo198214 - 05/24/2010
(05/24/2010, 04:41 AM)sheldonison Wrote:(05/23/2010, 07:54 AM)bo198214 Wrote: The problem with alternative fixed points is that we dont have a sickel between them. I.e. if we connect two conjugated non-primary fixed points with a straight line, then the image under \( e^z \) of this straight line overlaps itself. This is due to the imaginary part of the fixed point pair is apart more then \( 2\pi \) which makes the image of any connecting line between these fixed points, revolve around 0 at least once.Would this refer to the graph of the complex super-function, generated from the secondary fixed point, or to the graph of the real valued super-exponential, after the Kneser solution? Oh I refer here to the Kneser solution (and probably to the Kouznetsov solution too). These are the real-valued ones. It depends on an initial region bounded by some line between two fixed points and its image. Quote:I'd like to graph the 3*pi*i contour line of the super-function generated from the secondary fixed point, 2.0622777296+i*7.5886311785 By which method? I am also not really sure what you mean by 3*pi*i contour line of the super-function ... Quote:It is straightforward to generate the super-function from the secondary fixed point,You mean here regular iteration, giving non-real values on the real line? RE: tetration from alternative fixed point - sheldonison - 05/24/2010
(05/24/2010, 11:43 AM)bo198214 Wrote:Yes, by superfunction, I mean regular iteration, complex valued at the real axis, which is connected to the sexp by the \( \theta \) mapping function.(05/24/2010, 04:41 AM)sheldonison Wrote: Would this refer to the graph of the complex super-function, generated from the secondary fixed point, or to the graph of the real valued super-exponential, after the Kneser solution? \( \text{sexp}_e(z)=\text{superfunc}_e(z+\theta(z)) \) The superfunction developed from the secondary fixed point has an imaginary contour line, where the imaginary value of z=3*pi*i. My hypothesis is that the real values of z on this contour line would range from +infinity to -infinity. This is analogous to the superfunction developed from the primary fixed point, which has an imaginary contour line where the imaginary value of z=pi*i, and where the real ranges from +infinity to -infinity. After the Riemann mapping this contour line is on the real axis, from -3 to -2. The exponent of this line segment is on the real axis from -2 to -1, with values from -infinity to zero. I do see that the secondary superfunction would lead to a sexp with a total of 6*pi*i windings around the singularity at -2. I'll post a picture of the 3*pi*i contour line when I get the arithmetic working, analogous to the picture I posted of the contour line from the primary fixed point, http://math.eretrandre.org/tetrationforum/showthread.php?tid=213&pid=4531&mode=threaded RE: tetration from alternative fixed point - sheldonison - 06/03/2010
(05/24/2010, 03:03 PM)sheldonison Wrote: ....I do see that the secondary superfunction would lead to a sexp with a total of 6*pi*i windings around the singularity at -2.I haven't gotten the contour picture yet (and that would be the contours of the complex valued superfunction before the Riemann mapping), but I wanted to point at that 6*pi*i windings (after the Riemann mapping, which I have virtually no hope of computing) around the singularity of sexp_secondary(z=-2) is consistent with the taylor series for the sexp_secondary(z) at z=-1 having a lowest power term of z^3. The first and second derivatives would both be zero at f(z=-1). So I still think it might work, and might be analytic, although perhaps not terribly interesting, since the graph would be very very wobbly, with an inflection point and a derivative of zero at integers>=-1, z=-1,0,1,2,3 ..... - Shel RE: tetration from alternative fixed point - sheldonison - 06/21/2010
(05/23/2010, 07:54 AM)bo198214 Wrote:Of course, Henryk is correct, although I doubt I understand it well enough to prove anything. I can make graphs of the real contour of the superfunction developed from the alternative fixed point, with singularities at 0, 1, e, e^e, e^e^e .... and with a repeating pattern, but there's a couple of major differences in the behavior of the singularity, as compared to the superfunction developed from the primary fixed point.(05/23/2010, 12:05 AM)sheldonison Wrote: Has anyone generated the base e tetration from an alternative fixed point? The second fixed point (as far as I can tell) for base e is 2.0622777296+i*7.5886311785 .... (1) there is a gap between consecutive iterations of the contour from the secondary fixed point. For the primary fixed point, the singularity smoothly transitions from one side of the singularity to the other side. If you make a line between the singularity generated from the inverse superfunction on either side of zero+/- delta, the superfunction of that line converges nicely to zero. But this doesn't work for the secondary fixed point! (2) The inverse superfunction from the secondary fixed doesn't match the repeating real contour with singularities in it. For the inverse superfunction, I can only get the pattern for one iteration, between -inf and 0, or between 0 and 1, or between 1 and e. But I can't get more than one of them at a time. This is really the same problem as (1). I imagine the complete Riemann surface would somehow connect the different regions of the multi-valued inverse superfunction.... As a result of this, any Reimann mapping from the secondary fixed point will have discontinuities in the derivatives, at the integer values of Sexp(-1,0,1,2,3,4....). - Sheldon RE: tetration from alternative fixed point - bo198214 - 06/27/2010
(06/21/2010, 04:24 PM)sheldonison Wrote: I can make graphs of the real contour of the superfunction developed from the alternative fixed point, with singularities at 0, 1, e, e^e, e^e^e .... I think we need to go into the details here. If I compute the Abel function, I dont get singularities at 1,e,e^e,... for the secondary fixed point. I use something similar to the formula: \( \text{slog}(z)=\log_c(\log_\ast^{[n]}(z)-e[2])-n \) where c is the derivative at the secondary fixed point e[2] and \( \log_\ast \) is the branch of the logarithm with imaginary part between \( 2\pi \) and \( 4\pi \), which implies that \( \log_\ast^{[n]}(z)\to e[2] \). But with this formula repeated application of \( \log_\ast \) to 1 is possible without running into 0. So no singularities at 1,e,e^e,etc. How do you do it? RE: tetration from alternative fixed point - tommy1729 - 06/27/2010
(06/27/2010, 06:02 AM)bo198214 Wrote: If I compute the Abel function, I dont get singularities at 1,e,e^e,... for the secondary fixed point. is that slog(z) analytic ? is the related superfunction a superfunction of exp(z) + 2pi i then ? but then it cant be related to the superfunction of exp(z) , can it ? RE: tetration from alternative fixed point - bo198214 - 06/28/2010
(06/27/2010, 10:18 PM)tommy1729 Wrote: is that slog(z) analytic ?Yes its the regular slog at the secondary fixed point of exp. Quote:is the related superfunction a superfunction of exp(z) + 2pi i then ? No, its a superfunction of exp, even an entire one. Quote:but then it cant be related to the superfunction of exp(z) , can it ? its related via the Riemann mapping, see the Kneser thread for more details. |