08/22/2007, 11:28 PM

Well, I've prepared some graphs in the hope that I can address the question about my super-logarithm extension. The question was "are you sure?" about my super-logarithm definition not working for bases between . My only answer is to look at the graphs.

These are some graphs of the first three coefficients of the super-logarithm, i.e. the Abel function of . In each graph there are multiple curves, each curve corresponds to a specific approximation; the approximations shown are roughly and the independant axis represents the base . What follows are graphs of as functions of b for for the functional equation with .

As you can see, when the base is between , although the matrix equation has a solution, the solutions (coefficients) do not seem to converge. This is what leads me to beleive that my super-logarithm definition will only work for .

Andrew Robbins

These are some graphs of the first three coefficients of the super-logarithm, i.e. the Abel function of . In each graph there are multiple curves, each curve corresponds to a specific approximation; the approximations shown are roughly and the independant axis represents the base . What follows are graphs of as functions of b for for the functional equation with .

As you can see, when the base is between , although the matrix equation has a solution, the solutions (coefficients) do not seem to converge. This is what leads me to beleive that my super-logarithm definition will only work for .

Andrew Robbins