(06/22/2010, 09:15 PM)mike3 Wrote: Another question: is this intuitive iteration thing the same as what you get from solving the equations given by substituting a Taylor series with unknown coefficients for slog in \( \mathrm{slog}_b(b^z) = \mathrm{slog}_b(z) + 1 \) and solving the resulting system of linear equations?yes.
Quote: If so, then maybe one could also try coming at this from that direction: try plugging the coefficients for the regular slog into those equations and then seeing that they do not satisfy them (being linear, they have only one solution.Its an infinite linear equation system.
In general it can have no, exactly one, or infinitely many solutions.
However in our case we know it has infinitely many solutions as \( \theta(\text{slog}(z))+\text{slog}(z) \) is another solution, for every 1-periodic \( \theta \).
All these solutions will satisfy the infinite linear equation system, particularly the regular one.
The specific about the intuitive Abel function is that it is obtained by solving the infinite linear equation system as limit of square-truncations of the equation system.
