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10/13/2009, 03:53 AM
(This post was last modified: 10/13/2009, 08:07 PM by BaseAcid 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 tetrationpattern 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 largemodulus 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 tetralogarithm). 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!
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(10/13/2009, 03:53 AM)BaseAcid 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.
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10/13/2009, 07:46 PM
(This post was last modified: 10/13/2009, 09:47 PM by BaseAcid 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 nonreal fixed points?
Also to investigate:
will pentation for base e have any nontrivial singularities?
Basee pentation may be an entire function.
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10/13/2009, 09:47 PM
(This post was last modified: 10/15/2009, 09:41 PM by BaseAcid Tetration.)
BAT Wrote:Basee 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 nonconstant (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 nontrivial realtoreal nexponential 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 nonprinicipal branches of tetration (analytically continued around z = 2) in our construction of a holomorphic pentation? Too complex for me! [no pun intended]
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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)BaseAcid Tetration Wrote: BAT Wrote:Basee 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.
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(10/13/2009, 09:47 PM)BaseAcid 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).
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10/15/2009, 09:39 PM
(This post was last modified: 10/15/2009, 09:52 PM by BaseAcid 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)BaseAcid 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 nonprinicipal 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 logarithmictype 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?
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(10/15/2009, 09:39 PM)BaseAcid 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.
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Are the singularities of tetration branch points, poles or essential singularities?
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(10/16/2009, 08:52 PM)BaseAcid 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.
