In response to the thread Solving tetration for base 0 < b < e^-e. I will feature here what one could call regular superlogarithm.

The idea is the following: One can obtain the regular Abel function of by the Julia function :

(*)

The Julia function (or also called iterative logarithm) can be obtained by

(**)

where we mean to be the *regular* iteration of .

It is similar to the logarithm:

And we have a way to obtain the regular iteration by matrix powers.

Before we begin we reduce the problem of iterating to iterating . This is a linear conjugation with and , i.e. . It moves the fixed point from to .

has the powerseries coefficients:

Then we take the Carleman matrix of this series (I truncate it to 10 here):

And then we consider the first line of the matrix power . These are the coefficients of :

Each coefficient is a mixture of polynomials in containing . It has 0 convergence radius.

We apply (**) and get the coefficients of the Julia function :

Now we apply (*):

When we take the reciprocal of the julia function we get a formal Laurent series starting at -1. The coefficient at -1 is , the other coefficients are:

When integrating the coefficient of becomes the coefficient of the logarithm. The other coefficients of the Abel function are obtained by formal integration. The coefficients of the Abel function of are:

The superlogarithm is then just

.

Notes:

1. The provided values are just for comparison if you want to reproduce this approach. I worked with Sage and my own powerseries libary. I can provide more coefficients as numerics if you specify some format. I didnt test my computations against obvious errors nor plotted the corresponding graph, nor do I know whether the series converges (thoug it looks quite so).

2. The powers in the first row of the matrix power are of course ambigous. I choose the standard branch of the logarithm in .

3. In the formula for you can of course choose arbitrary cuts and branches for the logarithm. The corresponding cut is then the cut for the Abel function and the cut after application of is the cut of the superlogarithm in the point .

4. You can add arbitrary constants to the superlogarithm and obtain again a superlogarithm.

The idea is the following: One can obtain the regular Abel function of by the Julia function :

(*)

The Julia function (or also called iterative logarithm) can be obtained by

(**)

where we mean to be the *regular* iteration of .

It is similar to the logarithm:

And we have a way to obtain the regular iteration by matrix powers.

Before we begin we reduce the problem of iterating to iterating . This is a linear conjugation with and , i.e. . It moves the fixed point from to .

has the powerseries coefficients:

Code:

`0, -1, 1/2, -1/6, 1/24, -1/120, 1/720, -1/5040, 1/40320, -1/362880, ...`

Then we take the Carleman matrix of this series (I truncate it to 10 here):

And then we consider the first line of the matrix power . These are the coefficients of :

Each coefficient is a mixture of polynomials in containing . It has 0 convergence radius.

We apply (**) and get the coefficients of the Julia function :

Code:

`0, I*pi, -1/4*I*pi, 1/8*I*pi - 1/12, -7/96*I*pi + 1/48, 3/64*I*pi - 1/80, -41/1280*I*pi + 17/2880, 263/11520*I*pi - 41/12096, -901/53760*I*pi + 229/120960, 3245/258048*I*pi - 227/207360`

Now we apply (*):

When we take the reciprocal of the julia function we get a formal Laurent series starting at -1. The coefficient at -1 is , the other coefficients are:

When integrating the coefficient of becomes the coefficient of the logarithm. The other coefficients of the Abel function are obtained by formal integration. The coefficients of the Abel function of are:

The superlogarithm is then just

.

Notes:

1. The provided values are just for comparison if you want to reproduce this approach. I worked with Sage and my own powerseries libary. I can provide more coefficients as numerics if you specify some format. I didnt test my computations against obvious errors nor plotted the corresponding graph, nor do I know whether the series converges (thoug it looks quite so).

2. The powers in the first row of the matrix power are of course ambigous. I choose the standard branch of the logarithm in .

3. In the formula for you can of course choose arbitrary cuts and branches for the logarithm. The corresponding cut is then the cut for the Abel function and the cut after application of is the cut of the superlogarithm in the point .

4. You can add arbitrary constants to the superlogarithm and obtain again a superlogarithm.