Let us determine the regular super logarithm
of
,
at the lower fixed point
. Regular super logarithm shall mean that it satisfies
(1)=0)
(2)=\text{rslog}_b(x)+1)
and that
(3)
where the right side is the regular iteration of
at the fixed point
.
Then the formula for the principal Abel function is:
=\lim_{n\to\infty} \log_{\ln(a)}(a-\exp_b^{\circ n}(x))-n)
and that for the regular super logarithm:
= \alpha_b(x) - \alpha_b(1))
Graph of
:
Proof:
For doing this we first compute the regular Schroeder function (note that the Schroeder function is determined up to a multiplicative constant and the Abel function is determined up to an additive constant). A Schroeder function
of a function
is a function that satisfies the Schroeder equation
)=s\sigma(x))
We see that we can derive a solution
of the Abel equation
)=\alpha(x)+1)
by setting
.
Now there is the the so called principal Schroeder function
of a function
with fixed point 0 with slope
,
given by:
 = \lim_{n\to\infty} \frac{f^{\circ n}(x)}{s^n})
This function particularly yields the regular iteration at 0, via
.
To determine the Schroeder equation at the lower fixed point
of
we consider
with fixed point 0 and same slope
. Let
then
.
.
.
Hence
is the principial Schroeder function of
at
.
To get the principal Abel function we take the logarithm to base
:
=\sigma_f(a-x)=\lim_{n\to\infty} \frac{f^{\circ t}(a-x)}{s^n})=\lim_{n\to\infty} \frac{a-\exp_b^{\circ n}(a-(a-x))}{s^n}=\lim_{n\to\infty} \frac{a-\exp_b^{\circ n}(x)}{s^n})
.
(1)
(2)
and that
(3)
Then the formula for the principal Abel function is:
and that for the regular super logarithm:
Graph of
Proof:
For doing this we first compute the regular Schroeder function (note that the Schroeder function is determined up to a multiplicative constant and the Abel function is determined up to an additive constant). A Schroeder function
We see that we can derive a solution
by setting
Now there is the the so called principal Schroeder function
This function particularly yields the regular iteration at 0, via
To determine the Schroeder equation at the lower fixed point
Hence
To get the principal Abel function we take the logarithm to base