Let us determine the regular super logarithm of , at the lower fixed point . Regular super logarithm shall mean that it satisfies

(1)

(2)

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:

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 of a function is a function that satisfies the Schroeder equation

We see that we can derive a solution of the Abel equation

by setting .

Now there is the the so called principal Schroeder function of a function with fixed point 0 with slope , given by:

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 :

.

(1)

(2)

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:

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 of a function is a function that satisfies the Schroeder equation

We see that we can derive a solution of the Abel equation

by setting .

Now there is the the so called principal Schroeder function of a function with fixed point 0 with slope , given by:

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 :

.