Hi.
I noticed that the "gamma function" \( \Gamma(z) \), the continuous version of the factorial, i.e. continuum solution of
\( F(z+1) = z F(z) \)
in the complex \( z \)-plane, obeys the following law:
\( \lim_{t \rightarrow \infty} \Gamma(it) = 0 = \mathrm{fixpoint\ of\ recurrence} \)
similar to what Tetration does:
\( \lim_{t \rightarrow \infty} ^{it} e = \mathrm{fixpoint\ of\ \exp} \).
Is there something interesting here?
(and ditto for the lim to \( -\infty \))
Indeed, if you plot the two on graphs, they're kind-of similar to each other.
I also notice that the Gamma function can be uniquely specified by Wielandt's theorem, which uses the condition of being bounded in a strip parallel to the imag axis along with holomorphism on the right halfplane and the starting value). Now it seems this does not work entirely well for tetration, since here:
http://math.eretrandre.org/tetrationforu...452&page=2
an alternative strip-bounded solution, using different fixed points (but which is not as well-behaved, namely the inverse function has branch points at the real axis), was constructed. But I wonder whether the strip-bounding condition (plus holomorphism at at least the right half-plane and also that \( \mathrm{tet}(0) = 1 \)) at least determines the solution up to a possible choice of fixed points. Might it? As then the rest of the uniqueness would be simple: just add that the function should approach the two principal fixed points at \( \pm i\infty \), and then we'd have a full uniqueness specification for the tetrational (and this could probably also be generalized to functions defined via other recurrences as well).
I noticed that the "gamma function" \( \Gamma(z) \), the continuous version of the factorial, i.e. continuum solution of
\( F(z+1) = z F(z) \)
in the complex \( z \)-plane, obeys the following law:
\( \lim_{t \rightarrow \infty} \Gamma(it) = 0 = \mathrm{fixpoint\ of\ recurrence} \)
similar to what Tetration does:
\( \lim_{t \rightarrow \infty} ^{it} e = \mathrm{fixpoint\ of\ \exp} \).
Is there something interesting here?
(and ditto for the lim to \( -\infty \))
Indeed, if you plot the two on graphs, they're kind-of similar to each other.
I also notice that the Gamma function can be uniquely specified by Wielandt's theorem, which uses the condition of being bounded in a strip parallel to the imag axis along with holomorphism on the right halfplane and the starting value). Now it seems this does not work entirely well for tetration, since here:
http://math.eretrandre.org/tetrationforu...452&page=2
an alternative strip-bounded solution, using different fixed points (but which is not as well-behaved, namely the inverse function has branch points at the real axis), was constructed. But I wonder whether the strip-bounding condition (plus holomorphism at at least the right half-plane and also that \( \mathrm{tet}(0) = 1 \)) at least determines the solution up to a possible choice of fixed points. Might it? As then the rest of the uniqueness would be simple: just add that the function should approach the two principal fixed points at \( \pm i\infty \), and then we'd have a full uniqueness specification for the tetrational (and this could probably also be generalized to functions defined via other recurrences as well).