What are the types of complex iteration and tetration?

[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