What are the types of complex iteration and tetration?

[Daniel]

I'm pulling together a Mathematica notebook to submit to the Wolfram Function Repository and was thinking to name it ComplexIteration. With so many approaches here focused on real tetration I wonder if there are other approaches to complex iteration and tetration besides mine. Gottfried's work with matrices is the closest to my work that I know of, but then at least some approaches to real tetration start with employing complex tetration. All these approaches need to be consistent or derived from Schroeder's functional equation and the math is comparatively simple (for this forum), so I wouldn't be surprised if different approaches to complex tetration lead to the same place.

[Gottfried]

I'm pulling together a Mathematica notebook to submit to the Wolfram Function Repository and was thinking to name it ComplexIteration. With so many approaches here focused on real tetration I wonder if there are other approaches to complex iteration and tetration besides mine. Gottfried's work with matrices is the closest to my work that I know of, but then at least some approaches to real tetration start with employing complex tetration. All these approaches need to be consistent or derived from Schroeder's functional equation and the math is comparatively simple (for this forum), so I wouldn't be surprised if different approaches to complex tetration lead to the same place.

Hi Daniel -

 two comments.

1) I'm looking (again, currently) at the idea, whether the real Kneser-solution might be understandable as the limit of the matrix-solution (which I call the "polynomial" method) when the matrix-size /(n \times n \) is extrapolated to \( n \to \infty \). This simple method, for the truncated Carlemanmatrices, is real for real heights and real x - but of course has distortion against the Kneser-method. However, this distortion seems to diminuish with increasing matrixsize, so for 2nx2n instead of nxn I got 2 or 3 more digits towards the Kneser solution for some bases \( b \) in \( b^x \) . I called this one time the "poor man's Kneser interpolation", but knowing that I have no proof (and even no idea of it) for the asymptotic. But *if* this would come out, then we have a second real-to-real solutions which is not derived from the complex (Schroeder) method (and also not of Andy's method -as well real-to-real, but possibly converging to another interpolation)   

2) For some more educational/didactical introductions one might look at various methods of interpolation of the coefficients of the formal powerseries. We list the coefficients of formal powerseries for the zeroth, first, second, third,... iteration of the function and try to find meaningful interpolations: "polynomial" (if this is working), "exponential polynomial" (as I christened this once), and possibly others, which might as well arrive at real-to-real solutions. But if I recall correctly my explorations of this were based on iteration of \( \exp(z)-1 \) and \( t^z -1 \) so this is likely not what you are interested in.  

Just some (nearly) random thoughts, can't currently not dive in much deeper.

Gottfried

[Daniel]

I'm pulling together a Mathematica notebook to submit to the Wolfram Function Repository and was thinking to name it ComplexIteration. With so many approaches here focused on real tetration I wonder if there are other approaches to complex iteration and tetration besides mine. Gottfried's work with matrices is the closest to my work that I know of, but then at least some approaches to real tetration start with employing complex tetration. All these approaches need to be consistent or derived from Schroeder's functional equation and the math is comparatively simple (for this forum), so I wouldn't be surprised if different approaches to complex tetration lead to the same place.

Hi Daniel -

 two comments.

1) I'm looking (again, currently) at the idea, whether the real Kneser-solution might be understandable as the limit of the matrix-solution (which I call the "polynomial" method) when the matrix-size /(n \times n \) is extrapolated to \( n \to \infty \). This simple method, for the truncated Carlemanmatrices, is real for real heights and real x - but of course has distortion against the Kneser-method. However, this distortion seems to diminuish with increasing matrixsize, so for 2nx2n instead of nxn I got 2 or 3 more digits towards the Kneser solution for some bases \( b \) in \( b^x \) . I called this one time the "poor man's Kneser interpolation", but knowing that I have no proof (and even no idea of it) for the asymptotic. But *if* this would come out, then we have a second real-to-real solutions which is not derived from the complex (Schroeder) method (and also not of Andy's method -as well real-to-real, but possibly converging to another interpolation)   

2) For some more educational/didactical introductions one might look at various methods of interpolation of the coefficients of the formal powerseries. We list the coefficients of formal powerseries for the zeroth, first, second, third,... iteration of the function and try to find meaningful interpolations: "polynomial" (if this is working), "exponential polynomial" (as I christened this once), and possibly others, which might as well arrive at real-to-real solutions. But if I recall correctly my explorations of this were based on iteration of \( \exp(z)-1 \) and \( t^z -1 \) so this is likely not what you are interested in.  

Just some (nearly) random thoughts, can't currently not dive in much deeper.

Gottfried

Thanks Gottfried, your infinite matrix method reminds me of my idea that complex tetration becomes real tetration when a fixed point at infinity is used.

I was reviewing the latest version of Mathematica and it supports fractional differentiation. There is also a Carleman matrix notebook in Wolfram Function. I'm playing with the idea of a dynamical system based on differentiation that are convergent - have a fixed point at 0.

[JmsNxn]

...complex tetration becomes real tetration when a fixed point at infinity is used.

This is probably absolutely true. And in fact, can probably be derived from Paulsen's paper; and the uniqueness he described.

Essentially if:

\[
F(\pm i \infty) = L^{\pm}\\
\]

And \(F(z) = e^{F(z+1)}\), then \(F(z) = \text{tet}_K(z + z_0)\) for some constant \(z_0\). This should be true. But you may need to tighten your conditions a good amount; particulary that \(\overline{F(\pm i \infty + z)} = F(\mp i \infty + \overline{z})\)--and does so with good asymptotic data. (You would need some knowledge about how it approaches the fixed point at infinity).

PS: Super cool you're sending off a package to Wolfram

[Daniel]

For using complex tetration with a fixed point at imaginary infinity consider that the further the fixed point is from the real axis, the greater the dampening of the complex oscillations. But the dynamics of the complex tetration with a fixed point at imaginary infinity must be the same as that of the next further out fixed point. Since the further fixed point must dampen the oscillations even more, and yet be the same as the previous fixed point, the only feasible scenario is if the oscillations are complexly dampened out and we are looking at real tetration. Unfortunately I'm having troubles thinking of a constructive approach that provides actual values.

[JmsNxn]

OH I APOLOGIZE DANIEL

I thought you meant tetration at imaginary infinity decays to a fixed point of the exponential. You mean, at imaginary infinity it tends to imaginary infinity. Just how \(^z \eta\) behaves. This is an open problem. I don't know if it's solvable. But, it would be one helluva theorem and need one helluva proof to pull off.