08/07/2007, 04:38 PM
I just read Andrew Robbins' solution to the tetration problem, which I find very convincing, and want to use the opportunity to present and discuss it here.
The solution
satisfies the 2 natural conditions
1.
and 
2.
is infinitely differentiable.
However for avoiding difficulties with a later expansion, he instead solves for the tetration logarithm tlog (which he calls super logarithm but I find "tetration logarithm" somewhat more specific), which is the inverse of
, i.e.
.
The first condition is then translated into
1'.
and =\text{tlog}_b(b^x)-1)
while the second condition is equivalent to that also
2'.
is infinitely differentiable.
By 1' we merely need to consider the tlog on the interval
and can then derive the values for the other intervals
,
,
etc. above and
below.
The idea is now that we define a smooth (infinitely differentiable) function t on
and ensure that the by 1' resulting function is also smooth at the joining points of the intervals. Obviously it suffices to ensure this for the joining point 0 (and by 1' this transposes to each other joining point).
We simply expand t into a power series at 0 and then try to determine the coefficients such that the resulting function is smooth at 0
 = \sum_{i=0}^\infty \nu_i \frac{x^i}{i!})
is the i-th derivative of t at 0.
The resulting function on
is
=t(b^x)-1 )
We have to ensure that
for each
. What is now the k-th derivative of s at 0?
For
we get
.
For
the constant -1 vanishes and we make the following calculations:
}(x)=(t(b^x))^{(k)}=\left(\sum_{i=0}^\infty \nu_i \frac{b^{xi}}{i!}\right)^{(k)}=\sum_{i=0}^\infty\frac{\nu_i}{i!}(b^{xi})^{(k)})
The derivation of
is easily determined to be
and so the k-th derivative is
, which give us in turn
for
.
This is an infinite linear equation system system in the variables
.
The way of Andrew Robbins is now to approximate a solution by considering finite linear equation systems consisting of n equations and n variables
resulting from letting
for
.
First one can show that these equation systems have a unique solution for b>1 and numerical evidence then shows that
converges and that the resulting
are a solution of the infinite equation system.
Further numerical evidence shows, that the infinite sum in the definition of t converges for the so obtained
.
However I would guess that the claimed uniqueness for a solution satisfying 1' and 2' is not guarantied. We can use different approximations, for example for a given constant we can consider the equation systems, resulting from letting
for
. Because interestingly the sum
converges to
where e is the Euler constant and B_k are the Bell numbers. So by setting
for
the remaining sum
converges and merely introduces an additive constant in the linear equation system. The obtained solutions are different from the solution obtained by c=0.
However I didnt verify yet the convergence properties of these alternative solutions.
The solution
1.
2.
However for avoiding difficulties with a later expansion, he instead solves for the tetration logarithm tlog (which he calls super logarithm but I find "tetration logarithm" somewhat more specific), which is the inverse of
The first condition is then translated into
1'.
while the second condition is equivalent to that also
2'.
By 1' we merely need to consider the tlog on the interval
The idea is now that we define a smooth (infinitely differentiable) function t on
We simply expand t into a power series at 0 and then try to determine the coefficients such that the resulting function is smooth at 0
The resulting function on
We have to ensure that
For
For
The derivation of
This is an infinite linear equation system system in the variables
The way of Andrew Robbins is now to approximate a solution by considering finite linear equation systems consisting of n equations and n variables
First one can show that these equation systems have a unique solution for b>1 and numerical evidence then shows that
Further numerical evidence shows, that the infinite sum in the definition of t converges for the so obtained
However I would guess that the claimed uniqueness for a solution satisfying 1' and 2' is not guarantied. We can use different approximations, for example for a given constant we can consider the equation systems, resulting from letting
However I didnt verify yet the convergence properties of these alternative solutions.