12/09/2007, 12:27 AM
There are essentially 3 overlapping domains over which tetration is well-defined. These domains are of the form \( B \times A \) where B is the set of bases and A is the set of heights over which tetration is defined. But for the sake of simplicity, we will only be talking about B since it is more interesting.
These three regions are obviously overlapping, since the set of regular bases form a subset of the complex plane, and the integers form a subset of the complex plane as well, so the first two are overlapping. The regular bases include \( [e^{-e}, e^{1/e}] \) and the natural bases include \( (1, \infty) \), so the intersection of these is \( (1, e^{1/e}] \), so the second two have some overlap as well. If we were to take all of these domains and being them together, the union of all of these is:
In the picture below, the horizontal is the real axis, the vertical is the imaginary axis, red means the regular region, and blue means the natural region. These are the set of all b such that \( {}^{a}b \) is well-defined for all complex a (it is possible to define tetration outside this region with either method, but who knows what discontinuities might happen):
![[Image: SuperLogBases.png]](http://tetration.itgo.com/up/SuperLogBases.png)
The region in the complex plane over which regular tetration is well-defined is the region in which \( |\exp_b'(h)| \le 1 \) (where h is the fixed point) if it is less than 1 it is known as the (open) hyperbolic region, and if it is equal to 1 it is known as the parabolic region, and union of these two regions is known as the (closed) regular region.
The region in the complex plane over which natural tetration is well-defined is unknown. However, I have been able to approximate the region experimentally. This region is given above as \( B_N \). This region is conservative, and the region over which natural tetration actually works may be larger than this (for example the 2 might be closer to 1).
The unions of these two regions covers all of the positive real line except for \( (0, e^{-e}) \) but before we get disappointed, I have a comment. Outside the regular domain, regular tetration is still defined and calculable, but gives complex values for the real line. It is this that makes us avoid it in preference to natural tetration on the real line above eta. Since regular tetration gives complex values for non-integer heights where the base is in \( [e^{-e}, 1) \), it matters less that regular tetration gives complex values in \( (0, e^{-e}) \) since it makes all bases in (0, 1) complex valued. So this brings up an interesting point. There are essentially three kinds of domains we should be considering for each method of calculating non-integer tetration:
Another thing to note about these sets is the following. Where method X can be defined is completely determined, and can be as large as the complex plane, subject to the conditions of the method. The domain over which method X corresponds to tetration is entirely up to us, as we are in the process of defining tetration.
Maybe since we have a method that works well for complex numbers (regular), and a method that works well for real numbers (natural), we could use the natural method for bases near the positive real axis, and use the regular method for all other complex bases, despite the fact that the bases are outside the "regular domain". The nice thing about this is that although it might be discontinuous between the "regular" and "natural" methods, it defines tetration for all complex bases, so tetration can become a piecewise-holomorphic function:
with 2 pieces as opposed to an infinite number of pieces in the linear approximation of tetration.
Andrew Robbins
- The integer domain (complex bases, integer height)
- The regular domain (regular bases, complex height)
- The natural domain (natural bases, complex height)
These three regions are obviously overlapping, since the set of regular bases form a subset of the complex plane, and the integers form a subset of the complex plane as well, so the first two are overlapping. The regular bases include \( [e^{-e}, e^{1/e}] \) and the natural bases include \( (1, \infty) \), so the intersection of these is \( (1, e^{1/e}] \), so the second two have some overlap as well. If we were to take all of these domains and being them together, the union of all of these is:
\( \begin{tabular}{rl}
B_R & = \{ b \text{ where } |\log({}^{\infty}b)| \le 1 \} \\
B_N & \approx \{ b \text{ where } \text{Re}(b) - 2|\text{Im}(b)| > 1 \} \\
B \times A & = \{(b, a) \text{ where } \left\{
\begin{tabular}{ll}
a \in \bb{C} & \text{if } b \in (B_R \cup B_N) \\
a \in \bb{Z} & \text{otherwise }
\end{tabular}\right\} \text{ for all } b \in \bb{C} \}
\end{tabular}
\)
B_R & = \{ b \text{ where } |\log({}^{\infty}b)| \le 1 \} \\
B_N & \approx \{ b \text{ where } \text{Re}(b) - 2|\text{Im}(b)| > 1 \} \\
B \times A & = \{(b, a) \text{ where } \left\{
\begin{tabular}{ll}
a \in \bb{C} & \text{if } b \in (B_R \cup B_N) \\
a \in \bb{Z} & \text{otherwise }
\end{tabular}\right\} \text{ for all } b \in \bb{C} \}
\end{tabular}
\)
In the picture below, the horizontal is the real axis, the vertical is the imaginary axis, red means the regular region, and blue means the natural region. These are the set of all b such that \( {}^{a}b \) is well-defined for all complex a (it is possible to define tetration outside this region with either method, but who knows what discontinuities might happen):
The region in the complex plane over which regular tetration is well-defined is the region in which \( |\exp_b'(h)| \le 1 \) (where h is the fixed point) if it is less than 1 it is known as the (open) hyperbolic region, and if it is equal to 1 it is known as the parabolic region, and union of these two regions is known as the (closed) regular region.
The region in the complex plane over which natural tetration is well-defined is unknown. However, I have been able to approximate the region experimentally. This region is given above as \( B_N \). This region is conservative, and the region over which natural tetration actually works may be larger than this (for example the 2 might be closer to 1).
The unions of these two regions covers all of the positive real line except for \( (0, e^{-e}) \) but before we get disappointed, I have a comment. Outside the regular domain, regular tetration is still defined and calculable, but gives complex values for the real line. It is this that makes us avoid it in preference to natural tetration on the real line above eta. Since regular tetration gives complex values for non-integer heights where the base is in \( [e^{-e}, 1) \), it matters less that regular tetration gives complex values in \( (0, e^{-e}) \) since it makes all bases in (0, 1) complex valued. So this brings up an interesting point. There are essentially three kinds of domains we should be considering for each method of calculating non-integer tetration:
- Over what domain is the method be well-defined? (objective, the red/blue regions)
- Over what domain can the method be defined? (objective, most of the complex plane)
- Over what domain does the method give tetration? (subjective)
- The Abel matrix is invertible.
- The natural Abel function exists. (coefficient convergence)
- The natural Abel function is analytic. (series convergence)
Another thing to note about these sets is the following. Where method X can be defined is completely determined, and can be as large as the complex plane, subject to the conditions of the method. The domain over which method X corresponds to tetration is entirely up to us, as we are in the process of defining tetration.
Maybe since we have a method that works well for complex numbers (regular), and a method that works well for real numbers (natural), we could use the natural method for bases near the positive real axis, and use the regular method for all other complex bases, despite the fact that the bases are outside the "regular domain". The nice thing about this is that although it might be discontinuous between the "regular" and "natural" methods, it defines tetration for all complex bases, so tetration can become a piecewise-holomorphic function:
\( {}^{a}b\text{ : } \bb{C} \ \times\ \bb{C} \rightarrow \bb{C} \)
with 2 pieces as opposed to an infinite number of pieces in the linear approximation of tetration.
Andrew Robbins
