05/21/2008, 06:00 PM

Jabotinsky and also Ecalle define something they call iterative logarithm (which I here abbreviate as ilog). It is defined by

Jabotinsky [1] mentiones the similarity to the convetional log:

And we can easily derive the equation by:

I also can verify this via my powerseries package for the example .

However Jabotinsky also claims that:

which I can neither derive nor which is confirmed by the powerseries package. When I set and

then

[1] Eri Jabotinsky, Analytic Iteration, Transactions of the American Mathematical Society, Vol. 108, No. 3 (Sep., 1963), pp. 457-477

Jabotinsky [1] mentiones the similarity to the convetional log:

And we can easily derive the equation by:

I also can verify this via my powerseries package for the example .

However Jabotinsky also claims that:

which I can neither derive nor which is confirmed by the powerseries package. When I set and

then

[1] Eri Jabotinsky, Analytic Iteration, Transactions of the American Mathematical Society, Vol. 108, No. 3 (Sep., 1963), pp. 457-477