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 Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 10/13/2009, 03:53 AM (This post was last modified: 10/13/2009, 08:07 PM by Base-Acid Tetration.) Now that we have the complex plot of tetration, let's look for the complex fixed points. I know we have the real fixed point at approx. -1.850354529027181418483437788. I don't have the proper precise tools to investigate the complex fixed points' existence and location with much detale. I am sorry to overburden you, but if someone posted calculation/complex plot of tet(z)-z it would be very nice, as we could see the fixed points as zeros of the function. If there are complex fixed points, I think the first pair will be somewhere near -W(-1) (+-conjugate). There probably are more fixed points at impractically large values of z with some specific ratio of real and imaginary parts. (the repeating tetration-pattern fractal roughly repeats at 3.something +- .something*i; the ratio of the real part vs. imaginary part will be relevant, as well as what the fractal (the large-modulus parts of tet(z)) is that repeats. to look for those large fixed points we may use tlog(tet(z)) - tlog(z) or, to the extent that tlog and tet cancel out, z - tlog(z) (tlog is tetra-logarithm). then we may tetrate the large zeros of z - tlog(z) to get the true value of the fixed points. Pentation is depending at us! bo198214 Administrator Posts: 1,412 Threads: 91 Joined: Aug 2007 10/13/2009, 07:45 AM (10/13/2009, 03:53 AM)Base-Acid Tetration Wrote: Now that we have the complex plot of tetration, let's look for the complex fixed points. You mean tetration base e? Though I think the complex fixed points are not so interesting for pentation. We have a real fixed point between -2 and -1 for all bases > 1. And also another real fixed point > 0 in the case b < e^{1/e}. These fixed points may be used to define pentation. Otherwise we may ask ourselfes why we define tetration for b< e^{1/e} at the real fixed point and not at complex fixed points. But maybe I will ponder about the topic some more. Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 10/13/2009, 07:46 PM (This post was last modified: 10/13/2009, 09:47 PM by Base-Acid Tetration.) (10/13/2009, 07:45 AM)bo198214 Wrote: You mean tetration base e? Though I think the complex fixed points are not so interesting for pentation. We have a real fixed point between -2 and -1 for all bases > 1. And also another real fixed point > 0 in the case b < e^{1/e}. But are we sure that there are no non-real fixed points? Also to investigate: will pentation for base e have any non-trivial singularities? Base-e pentation may be an entire function. Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 10/13/2009, 09:47 PM (This post was last modified: 10/15/2009, 09:41 PM by Base-Acid Tetration.) BAT Wrote:Base-e pentation may be an entire function. [unlike tetration]Shit, I was wrong! [textbook speak]We will now prove that pentation is not entire: Theorem. There exists no entire Pentation Pen_b(z), for b>1, such that Pen(0) = 1. Proof. Let Tet_b(z) be the principal branch of tetration. Pentation (big P) satisfies Pen(z+1)=Tet(Pen(z)) We know that for z <= -2, Tet_b(z) has a branch cut, and therefore is not defined/holomorphic. Suppose, contrary to our claim, that there exists an entire function Pen_b(z), such that pen_b(z+1)= Tet_b(Pen_b(z)). By Picard's Little, pen(z), being non-constant (Pen(1) = Tet(Pen(0)) = b), surjects to the (punctured at most once) complex plane. There exist values z for which Tet(z) <= -2, e.g. a portion of the interval (-2,1] where -infinity < Tet(z) < 0; so Pen(z) must not take on these values, for if Pen(z) were to have these values Pen(z+1) would not be defined. Since there exists more than one values, in fact an interval (infinitely many) of such values in that interval, (by continuity of tetration at (-2,1] and the intermediate value theorem), which the Pentation has values in, there must be places in the complex plane on which pentation is not holomorphic. (Let z0 be such a value for which Pen(z0) <= -2. It follows that Pen(z0 + 1) = Tet(Pen(z0)) is undefined, and Pentation is not holomorphic at z0 + 1.); contradiction found. Therefore any entire Pentation must be a constant, trivial Pentation (for which Pen_b(z) is equal to a fixed point of Tet_b(z)). Halmos. ("AM I RIGHT, BO???") It can be further proven that (1)there exists no entire non-trivial real-to-real n-exponential for n > 3; the proof is left to the reader as an exercise. [/textbook speak] Now WHERE are teh singularities/branch points of pentation? Or alternatively we can incorporate parts of non-prinicipal branches of tetration (analytically continued around z = -2) in our construction of a holomorphic pentation? Too complex for me! [no pun intended] andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 10/15/2009, 08:17 AM (This post was last modified: 10/15/2009, 08:18 AM by andydude.) (10/13/2009, 09:47 PM)Base-Acid Tetration Wrote: BAT Wrote:Base-e pentation may be an entire function. [unlike tetration]Shit, I was wrong![/quote] Tetration isn't entire, it's holomorphic over the complex plane cut along the negative reals < -2. bo198214 Administrator Posts: 1,412 Threads: 91 Joined: Aug 2007 10/15/2009, 05:01 PM (10/13/2009, 09:47 PM)Base-Acid Tetration Wrote: Theorem. There exists no entire pentation pen_b(z), for b>1, such that pen(0) = 1. Either you have to specify that pen must use the principal branch of tetration (the cut is nothing god given, its just a choice to our human preference; the real function lives on Riemann manifolds), or you have to modify your proof and use the singularities of tetration at integers <=-2 (and show before that indeed every tetration has singularities there). Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 10/15/2009, 09:39 PM (This post was last modified: 10/15/2009, 09:52 PM by Base-Acid Tetration.) (10/15/2009, 08:17 AM)andydude Wrote: Tetration isn't entire, it's holomorphic over the complex plane cut along the negative reals < -2.I said PENTation. (10/15/2009, 05:01 PM)bo198214 Wrote: (10/13/2009, 09:47 PM)Base-Acid Tetration Wrote: Theorem. There exists no entire pentation pen_b(z), for b>1, such that pen(0) = 1. Either you have to specify that pen must use the principal branch of tetration (the cut is nothing god given, its just a choice to our human preference; the real function lives on Riemann manifolds), or you have to modify your proof and use the singularities of tetration at integers <=-2 (and show before that indeed every tetration has singularities there). I DID say: BAT Wrote:Let Tet_b(z) be the principal branch of tetration.I fixed the proof to make this clearer. I also did say that if we didn't restrict ourselves to the principal branch an entire pentation may be possible: BAT Wrote:Or alternatively we can incorporate parts of non-prinicipal branches of tetration (analytically continued around z = -2) in our construction of a holomorphic pentation? I meant if it can just avoid -2, pentation can still be entire. Now the question is, is it morally good to restrict ourselves to the principal branch? (BTW, what kind of singularities does Tet have at integers<=-2? Are they all logarithmic-type branch points around which the function winds infinitely many times?) Let's get back to the point. WHERE THE HECK ARE THE COMPLEX FIXED POINTS? bo198214 Administrator Posts: 1,412 Threads: 91 Joined: Aug 2007 10/16/2009, 06:28 AM (10/15/2009, 09:39 PM)Base-Acid Tetration Wrote: I DID say: BAT Wrote:Let Tet_b(z) be the principal branch of tetration.I fixed the proof to make this clearer. I know that you said it *in the proof*. But its of no use there. If you have restricting conditions you have to mention them in the theorem, otherwise the theorem is false. Though I still think the theorem doesnt need this restriction. As I mentioned I think it is still true if you allow the continuation from above onto the ray (-2,-oo) (Like the logarithm is also defined on (0,-oo)) because there are still enough singularities on this ray that you can employ for your proof. Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 10/16/2009, 08:52 PM Are the singularities of tetration branch points, poles or essential singularities? andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 10/17/2009, 10:47 AM (10/16/2009, 08:52 PM)Base-Acid Tetration Wrote: Are the singularities of tetration branch points, poles or essential singularities? It is well known that there is a logarithmic singularity at -2, which is a specific kind of essential singularity. It is also known that there are essential singularities at -3, -4, etc, but exactly what kind of singularities these are is not well known. What we do know, is that if you exponentiate these singularities a certain number of times, then you get a logarithmic singularity, and if you exponentiate again, then you get zero. « Next Oldest | Next Newest »

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