Jabotinsky and also Ecalle define something they call iterative logarithm (which I here abbreviate as ilog). It is defined by
\( \text{ilog}(f)=\frac{\partial f^{\circ t}(x)}{\partial t}|_{t=0} \)
Jabotinsky [1] mentiones the similarity to the convetional log:
\( \frac{\partial b^t}{\partial t}|_{t=0}=b^t \ln(b)|_{t=0}=\ln(b) \)
And we can easily derive the equation \( \text{ilog}(f^{\circ w})=w\;\text{ilog}(f) \) by:
\( \frac{\partial \left(f^{\circ w}\right)^{\circ t}}{\partial t}|_{t=0}=
\frac{\partial f^{\circ w t}}{\partial t}|_{t=0}=
w\;\frac{\partial f^{\circ w t}}{\partial w t}|_{t=0}=w\;\text{ilog}(f) \)
I also can verify this via my powerseries package for the example \( f(x)=x+x^2+x^3 \).
However Jabotinsky also claims that:
\( \text{ilog}(f\circ g)=\text{ilog}(f)+\text{ilog}(g) \)
which I can neither derive nor which is confirmed by the powerseries package. When I set \( f(x)=x+x^2 \) and \( g(x)=x+x^2+x^3 \)
then
\( \text{ilog}(f\circ g)-(\text{ilog}(f)+\text{ilog}(g))=-\frac{1}{2}x^4+\frac{5}{2}x^5-\frac{22}{3}x^6+\frac{29}{2}x^7+\dots \)
[1] Eri Jabotinsky, Analytic Iteration, Transactions of the American Mathematical Society, Vol. 108, No. 3 (Sep., 1963), pp. 457-477
\( \text{ilog}(f)=\frac{\partial f^{\circ t}(x)}{\partial t}|_{t=0} \)
Jabotinsky [1] mentiones the similarity to the convetional log:
\( \frac{\partial b^t}{\partial t}|_{t=0}=b^t \ln(b)|_{t=0}=\ln(b) \)
And we can easily derive the equation \( \text{ilog}(f^{\circ w})=w\;\text{ilog}(f) \) by:
\( \frac{\partial \left(f^{\circ w}\right)^{\circ t}}{\partial t}|_{t=0}=
\frac{\partial f^{\circ w t}}{\partial t}|_{t=0}=
w\;\frac{\partial f^{\circ w t}}{\partial w t}|_{t=0}=w\;\text{ilog}(f) \)
I also can verify this via my powerseries package for the example \( f(x)=x+x^2+x^3 \).
However Jabotinsky also claims that:
\( \text{ilog}(f\circ g)=\text{ilog}(f)+\text{ilog}(g) \)
which I can neither derive nor which is confirmed by the powerseries package. When I set \( f(x)=x+x^2 \) and \( g(x)=x+x^2+x^3 \)
then
\( \text{ilog}(f\circ g)-(\text{ilog}(f)+\text{ilog}(g))=-\frac{1}{2}x^4+\frac{5}{2}x^5-\frac{22}{3}x^6+\frac{29}{2}x^7+\dots \)
[1] Eri Jabotinsky, Analytic Iteration, Transactions of the American Mathematical Society, Vol. 108, No. 3 (Sep., 1963), pp. 457-477
