Hi.

I found the following easy uniqueness theorem that characterizes the regular tetrational of the base \( b = \eta = e^{1/e} \), and perhaps also the whole regular tetrational (with attracting fixed point) (though base-\( \eta \) 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 \( F(z) \) satisfying

1. \( F(z + 1) = \eta^{F(z)} \)

2. \( F(0) = 1 \)

3. \( F(z) \) is holomorphic in the entire cut plane with real \( z \le -2 \) removed,

4. \( \lim_{r \rightarrow \infty} F(re^{i\theta}) = e \) for all \( \theta \ne (2n+1)\pi, n \in \mathbb{Z} \) (could be weakened simply to saying the limit exists)

Proof: We use what is called Carlson's theorem. Assume \( F(z) \) and \( G(z) \) are two different solutions of the above conditions. Then consider \( H(z) = F(z) - G(z) \). Carlson's theorem says if this function, in the right half-plane, vanishes at every nonnegative integer, and is bounded asymptotically by \( O(e^{u|z|}) \) for some \( u < \pi \), then it vanishes everywhere. Conditions 1 and 2 imply that \( F \) and \( G \) are equal at every nonnegative integer, thus \( H \) is zero there, and condition 4 implies the asymptotic bounding (if two functions \( f(x) \) and \( g(x) \) have a limit at a given point, then the difference \( f(x) - g(x) \) 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 \( H(z) = 0 \), so \( F(z) = G(z) \) 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 (\( \Re(z) \ge 0 \)) and condition 4 to the function being of exponential type of at most \( \pi \) in that same right half-plane. The modifications also provide the theorem characterizing the regular tetrationals for \( 1 < b < \eta \).

I found the following easy uniqueness theorem that characterizes the regular tetrational of the base \( b = \eta = e^{1/e} \), and perhaps also the whole regular tetrational (with attracting fixed point) (though base-\( \eta \) 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 \( F(z) \) satisfying

1. \( F(z + 1) = \eta^{F(z)} \)

2. \( F(0) = 1 \)

3. \( F(z) \) is holomorphic in the entire cut plane with real \( z \le -2 \) removed,

4. \( \lim_{r \rightarrow \infty} F(re^{i\theta}) = e \) for all \( \theta \ne (2n+1)\pi, n \in \mathbb{Z} \) (could be weakened simply to saying the limit exists)

Proof: We use what is called Carlson's theorem. Assume \( F(z) \) and \( G(z) \) are two different solutions of the above conditions. Then consider \( H(z) = F(z) - G(z) \). Carlson's theorem says if this function, in the right half-plane, vanishes at every nonnegative integer, and is bounded asymptotically by \( O(e^{u|z|}) \) for some \( u < \pi \), then it vanishes everywhere. Conditions 1 and 2 imply that \( F \) and \( G \) are equal at every nonnegative integer, thus \( H \) is zero there, and condition 4 implies the asymptotic bounding (if two functions \( f(x) \) and \( g(x) \) have a limit at a given point, then the difference \( f(x) - g(x) \) 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 \( H(z) = 0 \), so \( F(z) = G(z) \) 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 (\( \Re(z) \ge 0 \)) and condition 4 to the function being of exponential type of at most \( \pi \) in that same right half-plane. The modifications also provide the theorem characterizing the regular tetrationals for \( 1 < b < \eta \).