superexponential below -2

[GFR]

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Would that mean a possibility of general extension of x^x to negative x even with finite iterations ?

That is what we were looking for. Unfortunately, we found an infinbite set of complex infinities. Mmmm! By the way, also my provisional conclusion is only a conjecture, needing a precise demonstration, ... if possible.

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Then there could be also a possibility to define any a^x with a<0 as complex number. Some Web complex number calculators do it anyway.

Yes, this is possible and ... standard. Please see the attachment (Negbaseexp), showing an exponential at base b = -2, with:
- absolute values;
- real part;
- imaginary part;
- real and imaginary parts, "enveloped" into the + and - Abs values.

I am happy to be back discussing with you.

GFR