Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation

[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]

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]

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]

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]

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]

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]

Tetration isn't entire, it's holomorphic over the complex plane cut along the negative reals < -2.

I said PENTation.

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:

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:

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]

I DID say:

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]

Are the singularities of tetration branch points, poles or essential singularities?

[andydude]

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.