06/20/2008, 01:26 PM

There is an error in your calculation of the Abel function of dxp:

The regular Abel function (for a function with fixed point at 0) always has a singularity at 0. You can not expand it into a powerseries at 0.

In the hyperbolic case the Abel function is the logarithm of the Schroeder function (as Gottfried also pointed out in his post), here you can also see that it is not developable at 0, because log is not.

What you however can do is to express the Abel function as where M is a function, such that is holomorphic and so developable at at 0. This is an extension to holomorphic functions (so called meromorphic functions) which also allow a finite number of negative powers in the power series development, i.e.

.

I explained here how this comes.

To compute the inverse of a powerseries - which is a meromorphic function - we determine the first index such that then we divide by and get a powerseries with with . We can then compute the reciprocal of this powerseries and the inverse of is then

is a powerseries with negative powers.

In our case , and .

If we now integrate this:

Yes this is correct and this is equal to the rslog because of the following:

You put the formula

where is the lower fixed point, and .

If is the (principal) Abel function of we write this is as:

with as you already pointed out with .

Now the rslog computation is slightly different, we start with

i.e.

to show that both Abel functions of are identical we just need to show that

This is be done by the following equivalences:

And the last line is automatically satisfied by the regular iteration.

If you now redevelop

with the correct - note that the logarithmic term in is now developable at 0 - you see that the coefficients become infinite sums, which is again no finite description (except you find some closed expression).

andydude Wrote:We can now find the Abel function of dxp by integrating its Julia function, but we need to find the reciprocal first. Finding the reciprocal of a power series can be a tedious task, so I've done the work for you:

and since , we solve for by equating the coefficients of x, and the solution to these equations is:

The regular Abel function (for a function with fixed point at 0) always has a singularity at 0. You can not expand it into a powerseries at 0.

In the hyperbolic case the Abel function is the logarithm of the Schroeder function (as Gottfried also pointed out in his post), here you can also see that it is not developable at 0, because log is not.

What you however can do is to express the Abel function as where M is a function, such that is holomorphic and so developable at at 0. This is an extension to holomorphic functions (so called meromorphic functions) which also allow a finite number of negative powers in the power series development, i.e.

.

I explained here how this comes.

To compute the inverse of a powerseries - which is a meromorphic function - we determine the first index such that then we divide by and get a powerseries with with . We can then compute the reciprocal of this powerseries and the inverse of is then

is a powerseries with negative powers.

In our case , and .

If we now integrate this:

Quote:Lastly, we can relate these findings back to the super-logarithm.

Let be the Abel function of dxp.

Let .

Then is an Abel function of exp.

Proof.

. []

Yes this is correct and this is equal to the rslog because of the following:

You put the formula

where is the lower fixed point, and .

If is the (principal) Abel function of we write this is as:

with as you already pointed out with .

Now the rslog computation is slightly different, we start with

i.e.

to show that both Abel functions of are identical we just need to show that

This is be done by the following equivalences:

And the last line is automatically satisfied by the regular iteration.

Quote:This means that

If you now redevelop

with the correct - note that the logarithmic term in is now developable at 0 - you see that the coefficients become infinite sums, which is again no finite description (except you find some closed expression).