Hi -
using the notation \( x_1 = \exp_b^{^{oh_1}}(x_0) \) and \( x_2 = \exp_b^{^{oh_2}}(x_1) \) we do arithmetic in the height (or "iteration") parameter like \( x_2 = \exp_b^{^{oh_1+h_2}}(x_0) \)
What about infinite series instead of a sum?
If we have a sufficient method for continuous tetration, then, for instance we should get
\( x_2 = \exp_b^{^{o1/2+1/4+1/8+...}}(x_0) = \exp_b^{^{o1}}(x_0) = b^{x_0} \)
For "nice" bases 1<= b <=ß (= exp(exp(-1)) ) we should get even more; I don't see any reason, why the series in the height-parameter should not be allowed to diverge.
So, with base b=sqrt(2) the following expression
\( y = \exp_b^{^{o1 + 2 + 4+ 8 +...}}(x_0) \)
seems to make sense to me because the partial evaluations converge to y=2 , for instance if we choose x_0 = 1.
On the other hand, the analytical continuation for the geometric series with constant quotient q \( g(q) = 1+q+q^2+ ... \) at q=2 gives
\( g(2) = 1/(1-2) = -1 \)
But -substitued this into the height-parameter- then we should also have
\( y = 2 = \exp_b^{^{o -1 }}(x_0) = \log_b(x_0) = \log_b(1) = 0 \)
where we see a contradiction.
So it seems to me that tetration can restrict the validity/range-of-usability of analytic continuation (if we do not have another explanation/notion/way-of-handling of such effects).
---
In a second view, this extends to other functions similarly. We have another inequality if a divergent series is used in the exponent of the power-function, for instance, to base e:
\( y = e^{^{-1 -2 -4 -8 -16 - \dots }} = \frac1{e^1}*\frac1{e^2}*\frac1{e^4} * \dots = 0 \neq e^1 \)
Just another plot of meditations... <phew>
Gottfried
using the notation \( x_1 = \exp_b^{^{oh_1}}(x_0) \) and \( x_2 = \exp_b^{^{oh_2}}(x_1) \) we do arithmetic in the height (or "iteration") parameter like \( x_2 = \exp_b^{^{oh_1+h_2}}(x_0) \)
What about infinite series instead of a sum?
If we have a sufficient method for continuous tetration, then, for instance we should get
\( x_2 = \exp_b^{^{o1/2+1/4+1/8+...}}(x_0) = \exp_b^{^{o1}}(x_0) = b^{x_0} \)
For "nice" bases 1<= b <=ß (= exp(exp(-1)) ) we should get even more; I don't see any reason, why the series in the height-parameter should not be allowed to diverge.
So, with base b=sqrt(2) the following expression
\( y = \exp_b^{^{o1 + 2 + 4+ 8 +...}}(x_0) \)
seems to make sense to me because the partial evaluations converge to y=2 , for instance if we choose x_0 = 1.
On the other hand, the analytical continuation for the geometric series with constant quotient q \( g(q) = 1+q+q^2+ ... \) at q=2 gives
\( g(2) = 1/(1-2) = -1 \)
But -substitued this into the height-parameter- then we should also have
\( y = 2 = \exp_b^{^{o -1 }}(x_0) = \log_b(x_0) = \log_b(1) = 0 \)
where we see a contradiction.
So it seems to me that tetration can restrict the validity/range-of-usability of analytic continuation (if we do not have another explanation/notion/way-of-handling of such effects).
---
In a second view, this extends to other functions similarly. We have another inequality if a divergent series is used in the exponent of the power-function, for instance, to base e:
\( y = e^{^{-1 -2 -4 -8 -16 - \dots }} = \frac1{e^1}*\frac1{e^2}*\frac1{e^4} * \dots = 0 \neq e^1 \)
Just another plot of meditations... <phew>
Gottfried
Gottfried Helms, Kassel
