This post isn't to attempt to share some awesome new insight into how power series work. What I'm about to explain is probably well-known and has been for a century or more.
But it interests me. Before I started studying tetration a few months ago, I didn't know that a radius of convergence indicated a singularity. I mean, I was familiar with the idea that some power series had a radius of convergence, and obviously I was familiar with the idea of singularities. But at some point during my minimal formal calculus education (two years of high school calculus, a semester of vector calculus, and a semester of differential equations), my teachers and professors failed to make the connection explicit. And in my personal studies, including a foray into tensor math, I never came across it, probably because by that point, power series were more of an afterthought.
I also had very little experience with complex analysis until I started looking at the tetration problem. Funny too, because I figured that complex math wouldn't factor in: I assumed that tetration would at best be for real numbers.
So now I'm filling in holes in my knowledge, and I'm finding some fascinating things. The first was my discovery that, in the vicinity of the origin, and especially near the primary fixed points, Andrew's slog for base e approximates the sum of conjugate logarithms.
Then just today, I was thinking about the sexp function, having finally decided to turn my attention back to it a couple days ago. Anyway, I was thinking about the singularity at sexp(-2). It occurred to me that in the immediate vicinity of -2, it would look pretty much like)
, where 
is the coefficient for the first degree term in the power series for sexp. In fact, this would just reduce to +\ln(a_1))
.
As such, near the singularity, and assuming no other singularities in the immediate vicinity, the power series for sexp should approximately equal the power series for the natural logarithm.
Sure enough, I took my terms for the power series of the slog at 0, and calculated the reversion of the series to get the power series for the sexp (at -1). When I calculated the power series for the first derivative (a trivial calculation), I found that after the first half dozen terms, the terms of the first derivative of the sexp are alternating plus or minus 1 to within 1% or less, and by the 18th term, they're equal to +/- 1 to within 1 part in a million.
In other words, the terms of the power series of sexp(z-1) converge on the terms for the power series of ln(z+1).
For review, the terms for the power series of slog(z) converge on the terms of the power series of + \log_{\overline{c_0}}\left(z-\overline{c_0}\right))
, where 
is the primary fixed point for exponentiation (base e), and 
is its conjugate. This allowed me to develop an algorithm to speed convergence of Andrew's matrix solution for finding the power series of the slog.
What would be really nice is if we could do the same for the matrix solution for the sexp. Unfortunately, it's a non-linear equation, so I'm entirely sure that this new information helps. However, if one had an iterative solver that could handle the non-linearity, which needed only good approximations to start with, then this new information can certainly provide those good approximations.
At this point, I'm wondering if there is anything else that this new information can help me with. For example, I'm still analyzing the graph of the slog, and finding new and interesting facets to its fractal beauty. But I'm not really any closer to being able to use this information to derive a better way to solve for the slog or sexp. In other words, my accelerated solver is still the best tool I have for calculating as much accuracy as possible. More accuracy may be possible, but I don't know yet how to get it.
But it interests me. Before I started studying tetration a few months ago, I didn't know that a radius of convergence indicated a singularity. I mean, I was familiar with the idea that some power series had a radius of convergence, and obviously I was familiar with the idea of singularities. But at some point during my minimal formal calculus education (two years of high school calculus, a semester of vector calculus, and a semester of differential equations), my teachers and professors failed to make the connection explicit. And in my personal studies, including a foray into tensor math, I never came across it, probably because by that point, power series were more of an afterthought.
I also had very little experience with complex analysis until I started looking at the tetration problem. Funny too, because I figured that complex math wouldn't factor in: I assumed that tetration would at best be for real numbers.
So now I'm filling in holes in my knowledge, and I'm finding some fascinating things. The first was my discovery that, in the vicinity of the origin, and especially near the primary fixed points, Andrew's slog for base e approximates the sum of conjugate logarithms.
Then just today, I was thinking about the sexp function, having finally decided to turn my attention back to it a couple days ago. Anyway, I was thinking about the singularity at sexp(-2). It occurred to me that in the immediate vicinity of -2, it would look pretty much like
As such, near the singularity, and assuming no other singularities in the immediate vicinity, the power series for sexp should approximately equal the power series for the natural logarithm.
Sure enough, I took my terms for the power series of the slog at 0, and calculated the reversion of the series to get the power series for the sexp (at -1). When I calculated the power series for the first derivative (a trivial calculation), I found that after the first half dozen terms, the terms of the first derivative of the sexp are alternating plus or minus 1 to within 1% or less, and by the 18th term, they're equal to +/- 1 to within 1 part in a million.
In other words, the terms of the power series of sexp(z-1) converge on the terms for the power series of ln(z+1).
For review, the terms for the power series of slog(z) converge on the terms of the power series of
What would be really nice is if we could do the same for the matrix solution for the sexp. Unfortunately, it's a non-linear equation, so I'm entirely sure that this new information helps. However, if one had an iterative solver that could handle the non-linearity, which needed only good approximations to start with, then this new information can certainly provide those good approximations.
At this point, I'm wondering if there is anything else that this new information can help me with. For example, I'm still analyzing the graph of the slog, and finding new and interesting facets to its fractal beauty. But I'm not really any closer to being able to use this information to derive a better way to solve for the slog or sexp. In other words, my accelerated solver is still the best tool I have for calculating as much accuracy as possible. More accuracy may be possible, but I don't know yet how to get it.
~ Jay Daniel Fox
