Bases for which Tetration is well-defined

[GFR]

Hi Andrew!

Concerning:

There are essentially 3 overlapping domains over which tetration is well-defined. These domains are of the form \( B \times A \) where B is the set of bases and A is the set of heights over which tetration is defined. But for the sake of simplicity, we will only be talking about B since it is more interesting.

• The integer domain (complex bases, integer height)
  
• The regular domain (regular bases, complex height)
  
• The natural domain (natural bases, complex height)
  

...
In the picture below, the horizontal is the real axis, the vertical is the imaginary axis, red means the regular region, and blue means the natural region. These are the set of all b such that \( {}^{a}b \) is well-defined for all complex a (it is possible to define tetration outside this region with either method, but who knows what discontinuities might happen):

Could you please, for my parsonal use, be kind enough to better precise what do you mean by the regular and natural bases. Thanks.

GFR