GFR,

I was thinking about Your idea that infinite tetration might be complex in bigger area and You must be perfectly right. It must be complex everywhere where branches of W and ln does not cancel out perfectly. So, in principle, it could be complex even over all real arguments, converging , oscillating, or diverging. I am not sure about negative arguments, though. Nor about complex arguments. Nor about tripleton (I forgot the name of those unalgebraic extensions of complex numbers) , quaternion or octonion or sedenion arguments.

Also, what type of operation is self root if it gives inverse of tetration? It is kind the only exactly defined unique root, meaning one x can have only one selfroot?

I was thinking about Your idea that infinite tetration might be complex in bigger area and You must be perfectly right. It must be complex everywhere where branches of W and ln does not cancel out perfectly. So, in principle, it could be complex even over all real arguments, converging , oscillating, or diverging. I am not sure about negative arguments, though. Nor about complex arguments. Nor about tripleton (I forgot the name of those unalgebraic extensions of complex numbers) , quaternion or octonion or sedenion arguments.

Also, what type of operation is self root if it gives inverse of tetration? It is kind the only exactly defined unique root, meaning one x can have only one selfroot?