09/12/2009, 02:00 AM
Hi.
I was wondering about the possibility of extension of the tetrational function \( ^{z} b \) to real and complex towers z (also called "heights", but I like "tower" because it rhymes with "power") for exotic bases in the range \( 0 < b < e^{-e} \), as well as at the point \( b = e^{-e} \). Most extensions to tetration seem to concentrate on bases greater than 1, however I'm interested in the range of bases less than 1 due to the exotic behavior of their integer tetration.
It seems that the tetration for bases \( e^{-e} < b < e^{1/e} \) can be defined via the regular iteration technique, and for \( b < 1 \) it yields complex values. The technique also seems to be applicable to all bases inside the convergent region of the complex plane (the "kidney-bean" shaped area of the "tetration fractal"), and when using the principal branch of the logarithm and Lambert function we get a "principal branch" of the tetrational in that region of bases with a branch cut going to the left from the base b = 1, where it is continuous from above. The bases on the interval \( e^{-e} < b < 1 \) lie along this cut line. When tetration of these bases to real towers is plotted, it forms a spiral pattern.
However, once one gets below \( e^{-e} \), there is no more attracting fixed point and \( ^n b \) for natural numbers n does not settle down as \( n \rightarrow \infty \), but instead oscillates between two values (solutions of \( b^{b^z} = z \) -- I'm not sure if there exists a closed-form solution for them like how is a closed form for the fixed points of the exponential via Lambert's function, though), i.e. it converges to a 2-cycle. Thus the regular method does not seem to be useful for extending to these bases (regular iteration down into the repelling fixed point is possible but it yields a function that is entire on the complex plane, and that is not good, as I'd expect tetration to have branch points/singularities at z = -2, -3, -4, -5, ...). Yet I'm really curious to know what would happen if this tetration were extended to real and complex values of z, as it doesn't seem much has been done with it before, and it's a sort of "last frontier" of the real bases. The behavior for \( e^{-e} < b < 1 \) is interesting enough, but those bases still converge on a fixed point.
I was curious about Ansus' formula and also Kouznetsov's Cauchy integral method, though it is possible (but by no means certain) that there may be additional singularities on the z-plane (additional to those at z = -2, -3, -4, -5, ...), as the bases in \( e^{-e} < b < 1 \) have additional singularities on the z-plane (in both the left and right half-planes) due to complex periodicity with period \( \frac{2\pi i}{\log(\log(F))} \) where \( F = -\frac{W(-\log(b))}{\log(b)} \) is the principal fixed point of the exponential), so choosing a good countour for the latter may be tricky, plus there are uncertainties in what the asymptotic behavior should be.
And by Ansus' formula, I mean the formula from the continuous product, given by
\( \mathrm{tet}'_b(x) = \log(b)^{x} \mathrm{tet}'_b(0) \prod_{k=0}^{x-1} \mathrm{tet}_b(k) \).
which can also be written as an iterative formula with an integral:
\( \mathrm{tet}_b(x) = \mathrm{tet}'_b(0) \int_{-1}^{x} \log(b)^x \prod_{x} f(x+1) dx \).
However I haven't had much luck in getting the iterative integral formula to stabilize/converge. For base e and an initial function (f(x) = 1) given as a truncated Taylor series it seemed to converge for a bit then diverged again (I could post the code if you want).
The thread these formulas come from is here:
http://math.eretrandre.org/tetrationforu...273&page=1
I was wondering about the possibility of extension of the tetrational function \( ^{z} b \) to real and complex towers z (also called "heights", but I like "tower" because it rhymes with "power") for exotic bases in the range \( 0 < b < e^{-e} \), as well as at the point \( b = e^{-e} \). Most extensions to tetration seem to concentrate on bases greater than 1, however I'm interested in the range of bases less than 1 due to the exotic behavior of their integer tetration.
It seems that the tetration for bases \( e^{-e} < b < e^{1/e} \) can be defined via the regular iteration technique, and for \( b < 1 \) it yields complex values. The technique also seems to be applicable to all bases inside the convergent region of the complex plane (the "kidney-bean" shaped area of the "tetration fractal"), and when using the principal branch of the logarithm and Lambert function we get a "principal branch" of the tetrational in that region of bases with a branch cut going to the left from the base b = 1, where it is continuous from above. The bases on the interval \( e^{-e} < b < 1 \) lie along this cut line. When tetration of these bases to real towers is plotted, it forms a spiral pattern.
However, once one gets below \( e^{-e} \), there is no more attracting fixed point and \( ^n b \) for natural numbers n does not settle down as \( n \rightarrow \infty \), but instead oscillates between two values (solutions of \( b^{b^z} = z \) -- I'm not sure if there exists a closed-form solution for them like how is a closed form for the fixed points of the exponential via Lambert's function, though), i.e. it converges to a 2-cycle. Thus the regular method does not seem to be useful for extending to these bases (regular iteration down into the repelling fixed point is possible but it yields a function that is entire on the complex plane, and that is not good, as I'd expect tetration to have branch points/singularities at z = -2, -3, -4, -5, ...). Yet I'm really curious to know what would happen if this tetration were extended to real and complex values of z, as it doesn't seem much has been done with it before, and it's a sort of "last frontier" of the real bases. The behavior for \( e^{-e} < b < 1 \) is interesting enough, but those bases still converge on a fixed point.
I was curious about Ansus' formula and also Kouznetsov's Cauchy integral method, though it is possible (but by no means certain) that there may be additional singularities on the z-plane (additional to those at z = -2, -3, -4, -5, ...), as the bases in \( e^{-e} < b < 1 \) have additional singularities on the z-plane (in both the left and right half-planes) due to complex periodicity with period \( \frac{2\pi i}{\log(\log(F))} \) where \( F = -\frac{W(-\log(b))}{\log(b)} \) is the principal fixed point of the exponential), so choosing a good countour for the latter may be tricky, plus there are uncertainties in what the asymptotic behavior should be.
And by Ansus' formula, I mean the formula from the continuous product, given by
\( \mathrm{tet}'_b(x) = \log(b)^{x} \mathrm{tet}'_b(0) \prod_{k=0}^{x-1} \mathrm{tet}_b(k) \).
which can also be written as an iterative formula with an integral:
\( \mathrm{tet}_b(x) = \mathrm{tet}'_b(0) \int_{-1}^{x} \log(b)^x \prod_{x} f(x+1) dx \).
However I haven't had much luck in getting the iterative integral formula to stabilize/converge. For base e and an initial function (f(x) = 1) given as a truncated Taylor series it seemed to converge for a bit then diverged again (I could post the code if you want).
The thread these formulas come from is here:
http://math.eretrandre.org/tetrationforu...273&page=1
