Complex Tetration

[mike3]

The algorithm has to calculate the equivalent of Kneser's Riemann mapping for each base, before it can generate results, so it won't give you results for different bases as easily as you might like.

I see. So, what about Robbin's method? I think there is a pseudo-code version of his method in his article, on the code section. Can it be implemented in PARI? How much freely can we extend Robin's method to complex numbers? Is there any convergence/divergence issues for Robin's method (complex numbers)? Sorry for asking so many question, actually I am working on a "function" related to the tetration and want to gain full computational access so that I can continue regularizing the function.

Is tetration well-defined for negative integers? I mean, what is 2^-3? According to the PARI code you created, it is, in the Kneser sense \( 7.69052232412789846 + 4.53236014182719380 \cdot i \). Is this a valid argument? Don't we have \( {}^{-x} a = \log_{a}(0) \) for positive integer x? Shouldn't we consider that tetration has poles at the negative integers?

Balarka
.

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

Mmh.... on the question of negative integer heights:

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

2. On other branches, there may be a finite value at those points. On these other branches, there will be many additional branch point singularities in the right half-plane.