Hi.

I found the following easy uniqueness theorem that characterizes the regular tetrational of the base , and perhaps also the whole regular tetrational (with attracting fixed point) (though base- is particularly interesting since it seems that both the regular and non-regular (i.e. Kneser's, etc.) method approach the same tetrational at this base.), with some modification (of condition 4). The conditions are very simple and easy.

Theorem: There is a unique complex function satisfying

1.

2.

3. is holomorphic in the entire cut plane with real removed,

4. for all (could be weakened simply to saying the limit exists)

Proof: We use what is called Carlson's theorem. Assume and are two different solutions of the above conditions. Then consider . Carlson's theorem says if this function, in the right half-plane, vanishes at every nonnegative integer, and is bounded asymptotically by for some , then it vanishes everywhere. Conditions 1 and 2 imply that and are equal at every nonnegative integer, thus is zero there, and condition 4 implies the asymptotic bounding (if two functions and have a limit at a given point, then the difference does as well) because every function decaying to a fixed value will be bounded in the asymptotic by any exponential (can give proof here if needed to fill this out.). Thus , so and we are done. QED.

See:

http://mathworld.wolfram.com/CarlsonsTheorem.html

Indeed, this says that condition 3 can be weakened to just holomorphism in the right half-plane () and condition 4 to the function being of exponential type of at most in that same right half-plane. The modifications also provide the theorem characterizing the regular tetrationals for .

I found the following easy uniqueness theorem that characterizes the regular tetrational of the base , and perhaps also the whole regular tetrational (with attracting fixed point) (though base- is particularly interesting since it seems that both the regular and non-regular (i.e. Kneser's, etc.) method approach the same tetrational at this base.), with some modification (of condition 4). The conditions are very simple and easy.

Theorem: There is a unique complex function satisfying

1.

2.

3. is holomorphic in the entire cut plane with real removed,

4. for all (could be weakened simply to saying the limit exists)

Proof: We use what is called Carlson's theorem. Assume and are two different solutions of the above conditions. Then consider . Carlson's theorem says if this function, in the right half-plane, vanishes at every nonnegative integer, and is bounded asymptotically by for some , then it vanishes everywhere. Conditions 1 and 2 imply that and are equal at every nonnegative integer, thus is zero there, and condition 4 implies the asymptotic bounding (if two functions and have a limit at a given point, then the difference does as well) because every function decaying to a fixed value will be bounded in the asymptotic by any exponential (can give proof here if needed to fill this out.). Thus , so and we are done. QED.

See:

http://mathworld.wolfram.com/CarlsonsTheorem.html

Indeed, this says that condition 3 can be weakened to just holomorphism in the right half-plane () and condition 4 to the function being of exponential type of at most in that same right half-plane. The modifications also provide the theorem characterizing the regular tetrationals for .