(01/13/2016, 08:24 AM)andydude Wrote: @marraco, @tommy

That certainly is a cool equation, even if it is easily provable.

@everyone

Also, I think I can express my earlier comment in different words now. Tetration is defined as the 1-initialized superfunction of exponentials. The previous functions discussed earilier are 3-initialized and 5-initialized, which makes them, not tetration, by definition. However, if there is an analytic continuation of the 1-initialized superfunction that overlaps with the 3-initialized superfunction, AND if on the overlap f(0) = 3, then they can be considered branches of the same function. But until that is proven, I don't think it's accurate to say that they're all "tetration". They are, however, iterated exponentials in the sense that they extend to non-integer n. And so I would probably write these functions as instead of saying that is a multivalued function that returns all three.

That makes sense, but on other side, we need to solve equations like (3>2, the asymptotic limit).

Similarly, when we solve (-1<0, the asymptotic limit), we do not say that is is a function different than exponentiation. We just extend the domain to complex numbers.

but this equation has real solutions, unless we consider the pair a new kind of number.

If we use a new kind of number, then, for the main branch, ; no more a real number.

I have the result, but I do not yet know how to get it.