Complex Tetration

[mike3]

Yes, I forgot about that. Robbins' method does not in fact work for every base.

I am not really concerned for the base. In my case, it is enough if it works for all the positive integer base. I am much more interested in complex (or perhaps real) heights. (A. Robbins is here too? Mein gott, this seems like a whole community of mathematicians!)

Then you'll probably like the Kneser method more than Robbins' one. Robbins' only works for a limited range of complex heights since it creates a Taylor series in the height, which is limited by the singularities (and also, quasi-limited by the area of very rapid growth).

tetration is not defined for negative-integer heights equal to -2 or less. It has branch points (not poles) at those heights.

Thanks. But why the code gives such results? Is it doing something like regularization for negative integers smaller than -1? What if we define those point by taking principle values of those (if possible)?

What commands are you using to give tetration? What commands did you use to get that value? As when I use it, PARI/GP returns a "*** log: zero argument in mplog" error -- which is what I'd expect, since there's a singularity. I use "init(2)" then "loop(<some number of loops>)" then "sexp(-3)". Perhaps you missed a command or used the wrong one?

I don't know of a code off-hand, but I suppose I could whip one up.

I terribly need some code like that since I don't have mathematica or maple. Please create one if possible.

I'll see if I can write one.

I'm curious: what exactly are you trying to do with this, anyway?