Hi.
I was wondering: what happens if one tries to extend the tetration to its other branches in the complex numbers, and build up the Riemann surface?
Consider, for example, the regular tetration for bases \( e^{-e} \lt b \lt e^{1/e} \). We get:
\( {}^z b = \exp^z_b(1) = \lim_{n \rightarrow \infty}\log^n_b\left(F\left(1 - \ln(b)^z\right) + \ln(b)^z \exp^n_b(1)\right) \)
where
\( F = -\frac{W(-\ln(b))}{\ln(b)} \)
is the fixed point or limit as the tower goes to infinity over the integers.
(see this thread)
Let us now consider what is happening with the above formula. The portion \( F\left(1 - \ln(b)^z\right) + \ln(b)^z \exp^n_b(1) \) is an entire function of z for any given n. The part that begins to introduce the branch points, then, is the repeated taking of logarithms \( \log_b \). Consider the removal of the first log by exponentiating both sides to base b. This causes the left to become \( {}^{z+1} b \), which has no singularity at \( z = -2 \). So we see the outermost log creates the singularity at \( z = -2 \), and we can then deevelop all branches obtainable from whirling around this singularity only by adding integer multiples of the period \( \omega \) of the base-b exponential, i.e. \( \omega = \frac{2\pi i}{\ln(b)} \). This allows us to get
\( \mathrm{tet}_b_{[s_0]}(z) = \lim_{n \rightarrow \infty}\log^n_b\left(F\left(1 - \ln(b)^z\right) + \ln(b)^z \exp^n_b(1)\right) + \omega s_0,\ s_0 \in \mathbb{Z} \)
for the continuation around the lowest (logarithmic) order (i.e. logarithmic) singularities. By extension of this idea, we see the next log in generates the singularities of the next higher logarithmic order (i.e. double logarithmic), and we can say
\( \mathrm{tet}_b_{[s_0, s_1]}(z) = \lim_{n \rightarrow \infty}\log_b\left(\log^{n-1}_b\left(F\left(1 - \ln(b)^z\right) + \ln(b)^z \exp^n_b(1)\right) + \omega s_1\right) + \omega s_0,\ s_0, s_1 \in \mathbb{Z} \)
We can continue on further, and we obtain, for a general finite integer sequence \( s_0, s_1, ..., s_k \), which I call a "branch code":
\( \mathrm{tet}_b_{[s_0, s_1, ..., s_k]}(z) = \lim_{n \rightarrow \infty}\log^n_b_{[s_0, s_1, ..., s_k]}\left(F\left(1 - \ln(b)^z\right) + \ln(b)^z \exp^n_b(1)\right) \)
where we define
\( \log^n_b_{[s_0, s_1, ..., s_k]}(z) = \begin{cases}
\log_b\left(\log_b\left(...\log_b\left(\log_b(z) + \omega s_{n-1}\right) + ...\right) + \omega s_1\right) + \omega s_0,\ \mathrm{if}\ n \le k+1 \\
\log_b\left(\log_b\left(...\log_b\left(\log_b\left(\log_b^{n-(k+1)}(z)\right) + \omega s_{k}\right) + ...\right) + \omega s_1\right) + \omega s_0,\ \mathrm{if}\ n \g k+1
\end{cases} \),
and \( \log_b \) are principal branches of the base-b complex logarithm. Thus the collection of all branches constructed this way for all finite sequences, I'd believe, would form the Riemann surface and the complete multivalued function for tetration to that base. Note that there are also likely branch points in the base, however continuation on that would mean going outside the regular iteration to other methods that we don't know (it's sort of like tetrating b = 0.04 to complex height) -- though I wonder if the continuum sum formula might be able to do it (seen my other thread about that?). Right now, though, I'm just focusing on the extension around the branchpoints in \( z \). It may be possible to think of the sequence \( [s_0, s_1, ..., s_k] \) as meaning "start at the principal branch, go and run \( s_0 \) times around \( z = -2 \), then go and run \( s_1 \) times around \( z = -3 \), and so on, finally running \( s_k \) times around \( z = -2 - k \)". The principal branch has code \( [0] \), or \( [0, 0] \), etc.
One thing of note here is that on some of these other branches, points that were singularities on the principal branch may not be so on these. Consider, for example, the branch \( [0, 1] \). When we make the value for \( z = -2 \), the next-to-outermost log has \( \omega \) added to it. As its value is zero (since it represents \( ^{z+1} b \) as mentioned before and \( ^{-1} b = 0 \) on the principal branch) we get the value here as actually \( \log_b(\omega) \) instead of being a singularity with undefined value. \( z = -3 \) is still a singularity, however (it has to be, as we ran around it once to get here.).
Graphs of individual branches did not seem very interesting. Many appear "flatter" than the principal branch, interestingly (except, of course, for those obtained as \( \mathrm{tet}_b_{[s_0]}(z) \) with varying \( s_0 \)). But what happens when you put them all together? Say we glued together all branches with 4-integer codes ranging from \( [-4, -4, -4, -4] \) to \( [4, 4, 4, 4] \) -- \( 9^4 \) or 6561 branches, and base \( b = \sqrt{2} \). What comes out? I don't have any program capable of graphing multivalued graphs, which are 3-dim (actually 4-dim, but projected to 3-dim by viewing the real and imag part separately) layered objects. And 6561 layers is a boatload of layers! Maybe it'd be easier to start with, say, \( [-2, -2, 0] \) to \( [2, 2, 0] \), which has \( 5^2 \) or 25 layers, or the somewhat harder \( [-1, -1, -1, -1, 0] \) to \( [1, 1, 1, 1, 0] \), which has \( 3^4 \) or 81 layers. Does anyone have anything they could use to take a crack at this? I'd love to see some pictures.
There is one more property I thought worth mentioning. The formulas are given for finite sequences of integers only. By the limit process, we could define it for infinite sequences, which would generate uncountably many new values. However I am not sure whether or not these points could be truly thought of as values of tetration, because there appear to be certain theorems (see this sci.math newsgroup posting and thread I had a few months ago) that say the analytic continuation of a complex function to a multivalued function must produce only countably many values: presumably, the resulting "Riemann surfaces" from that limit procedure would contain disconnected elements (all the uncountably many sheets we just added), and so could not be interpreted as the result of analytic continuation of \( \mathrm{tet}_b(z) \) in the \( z \)-parameter. However, even if these cannot be interpreted as proper values of the tetrational, the resulting object may still be mathematically interesting in some way...
I was wondering: what happens if one tries to extend the tetration to its other branches in the complex numbers, and build up the Riemann surface?
Consider, for example, the regular tetration for bases \( e^{-e} \lt b \lt e^{1/e} \). We get:
\( {}^z b = \exp^z_b(1) = \lim_{n \rightarrow \infty}\log^n_b\left(F\left(1 - \ln(b)^z\right) + \ln(b)^z \exp^n_b(1)\right) \)
where
\( F = -\frac{W(-\ln(b))}{\ln(b)} \)
is the fixed point or limit as the tower goes to infinity over the integers.
(see this thread)
Let us now consider what is happening with the above formula. The portion \( F\left(1 - \ln(b)^z\right) + \ln(b)^z \exp^n_b(1) \) is an entire function of z for any given n. The part that begins to introduce the branch points, then, is the repeated taking of logarithms \( \log_b \). Consider the removal of the first log by exponentiating both sides to base b. This causes the left to become \( {}^{z+1} b \), which has no singularity at \( z = -2 \). So we see the outermost log creates the singularity at \( z = -2 \), and we can then deevelop all branches obtainable from whirling around this singularity only by adding integer multiples of the period \( \omega \) of the base-b exponential, i.e. \( \omega = \frac{2\pi i}{\ln(b)} \). This allows us to get
\( \mathrm{tet}_b_{[s_0]}(z) = \lim_{n \rightarrow \infty}\log^n_b\left(F\left(1 - \ln(b)^z\right) + \ln(b)^z \exp^n_b(1)\right) + \omega s_0,\ s_0 \in \mathbb{Z} \)
for the continuation around the lowest (logarithmic) order (i.e. logarithmic) singularities. By extension of this idea, we see the next log in generates the singularities of the next higher logarithmic order (i.e. double logarithmic), and we can say
\( \mathrm{tet}_b_{[s_0, s_1]}(z) = \lim_{n \rightarrow \infty}\log_b\left(\log^{n-1}_b\left(F\left(1 - \ln(b)^z\right) + \ln(b)^z \exp^n_b(1)\right) + \omega s_1\right) + \omega s_0,\ s_0, s_1 \in \mathbb{Z} \)
We can continue on further, and we obtain, for a general finite integer sequence \( s_0, s_1, ..., s_k \), which I call a "branch code":
\( \mathrm{tet}_b_{[s_0, s_1, ..., s_k]}(z) = \lim_{n \rightarrow \infty}\log^n_b_{[s_0, s_1, ..., s_k]}\left(F\left(1 - \ln(b)^z\right) + \ln(b)^z \exp^n_b(1)\right) \)
where we define
\( \log^n_b_{[s_0, s_1, ..., s_k]}(z) = \begin{cases}
\log_b\left(\log_b\left(...\log_b\left(\log_b(z) + \omega s_{n-1}\right) + ...\right) + \omega s_1\right) + \omega s_0,\ \mathrm{if}\ n \le k+1 \\
\log_b\left(\log_b\left(...\log_b\left(\log_b\left(\log_b^{n-(k+1)}(z)\right) + \omega s_{k}\right) + ...\right) + \omega s_1\right) + \omega s_0,\ \mathrm{if}\ n \g k+1
\end{cases} \),
and \( \log_b \) are principal branches of the base-b complex logarithm. Thus the collection of all branches constructed this way for all finite sequences, I'd believe, would form the Riemann surface and the complete multivalued function for tetration to that base. Note that there are also likely branch points in the base, however continuation on that would mean going outside the regular iteration to other methods that we don't know (it's sort of like tetrating b = 0.04 to complex height) -- though I wonder if the continuum sum formula might be able to do it (seen my other thread about that?). Right now, though, I'm just focusing on the extension around the branchpoints in \( z \). It may be possible to think of the sequence \( [s_0, s_1, ..., s_k] \) as meaning "start at the principal branch, go and run \( s_0 \) times around \( z = -2 \), then go and run \( s_1 \) times around \( z = -3 \), and so on, finally running \( s_k \) times around \( z = -2 - k \)". The principal branch has code \( [0] \), or \( [0, 0] \), etc.
One thing of note here is that on some of these other branches, points that were singularities on the principal branch may not be so on these. Consider, for example, the branch \( [0, 1] \). When we make the value for \( z = -2 \), the next-to-outermost log has \( \omega \) added to it. As its value is zero (since it represents \( ^{z+1} b \) as mentioned before and \( ^{-1} b = 0 \) on the principal branch) we get the value here as actually \( \log_b(\omega) \) instead of being a singularity with undefined value. \( z = -3 \) is still a singularity, however (it has to be, as we ran around it once to get here.).
Graphs of individual branches did not seem very interesting. Many appear "flatter" than the principal branch, interestingly (except, of course, for those obtained as \( \mathrm{tet}_b_{[s_0]}(z) \) with varying \( s_0 \)). But what happens when you put them all together? Say we glued together all branches with 4-integer codes ranging from \( [-4, -4, -4, -4] \) to \( [4, 4, 4, 4] \) -- \( 9^4 \) or 6561 branches, and base \( b = \sqrt{2} \). What comes out? I don't have any program capable of graphing multivalued graphs, which are 3-dim (actually 4-dim, but projected to 3-dim by viewing the real and imag part separately) layered objects. And 6561 layers is a boatload of layers! Maybe it'd be easier to start with, say, \( [-2, -2, 0] \) to \( [2, 2, 0] \), which has \( 5^2 \) or 25 layers, or the somewhat harder \( [-1, -1, -1, -1, 0] \) to \( [1, 1, 1, 1, 0] \), which has \( 3^4 \) or 81 layers. Does anyone have anything they could use to take a crack at this? I'd love to see some pictures.
There is one more property I thought worth mentioning. The formulas are given for finite sequences of integers only. By the limit process, we could define it for infinite sequences, which would generate uncountably many new values. However I am not sure whether or not these points could be truly thought of as values of tetration, because there appear to be certain theorems (see this sci.math newsgroup posting and thread I had a few months ago) that say the analytic continuation of a complex function to a multivalued function must produce only countably many values: presumably, the resulting "Riemann surfaces" from that limit procedure would contain disconnected elements (all the uncountably many sheets we just added), and so could not be interpreted as the result of analytic continuation of \( \mathrm{tet}_b(z) \) in the \( z \)-parameter. However, even if these cannot be interpreted as proper values of the tetrational, the resulting object may still be mathematically interesting in some way...
