Observations on power series involving logarithmic singularities

[bo198214]

\( (B-I)*P_{\small F}\~{}=[1, 0, 0, ...]\~{} \)

This is exactly what Andrew's matrix does (with the exception of explicitly removing the first column; see below).

Yup. (and removing the lowest row, that it remains a square matrix).

And now I see exactly why it should work, assuming A) that the infinite system has a unique solution, and B) that the partial solutions converge on this unique solution as we increase the matrix size.

Unfortunately A) is wrong. We know already that there are infinite many solutions for the infinite equation system. I even gave a particular different (non-sinus based) solution here.
I call Andrew's way of solving this equation system by truncated approximation the natural solution.
Which of course works for any Abel equations (if it converges), as he also stated somewhere.

Though I havent thoroughly verified it, it looks indeed as if the solution is independent of the development point of the power series (which is currently at 0). We should check this.

However this is not the matrix operator method of Gottfried, which gives real iterates of the original function and can be considered as a generalization of the solution of the hyperbolic iteration with a fixed point via the Schroeder equation. We have there
\( f^{\circ t}(x) = \sigma^{-1}(c^t\sigma(x)) \), or directly \( f^{\circ t}=\sigma^{-1}\circ {\mu_c}^{\circ t} \circ \sigma \).

Translated into matrix form
\( F^t = \Sigma^{-1} {dV_c}^t \Sigma \).

And the good news is that nearly each matrix has such a decomposition with a diogonal matrix in the middle, however it is not necessarily of the form of powers of one number, nonetheless one can take real powers of it.