jaydfox Wrote:And the quick and easy predication is that, in addition to the singularities at 2 and 4 and their images at imaginary offsets, there would be additional singularities stretching out towards positive infinity, with a relatively calm strip between them, much like the ribs off the backbone of the base e slog. And, as with that slog, there would be additional branches off those branches, between each set of singularities, though without the alternating "logarithmic" and "exponential" types. I still need to create a decent graph for this, because as yet I've only been able to picture it in my mind.

By the way, the corrugations introduced by the cyclic shift are actually "desired", in the sense that they help create the singularities at the logarithms of the locations of other singularities.

In fact, I had started to make this connection earlier, but hadn't put it together until Henryk described this very effect of "corrugations".

To see this in action, let's go back to the slog for base e. It has singularities at equally spaced intervals from the two primary fixed points at 0.318 +/- 1.337i.

So, the value of the slog along the imaginary axis (the line with real part 0) is a simple, well-behaved, complex-valued function. The value along the line with real part -1 is even more well-behaved, and so on as we move to the left.

If we were to develop a complex fourier series along the imaginary axis, it should behave very nicely, and yet introduce singularities at the fixed points.

So far, this is already interesting, and I had figured out this much about a week ago. But what's more interesting to me are the singularities inside the logarithmic branches. After Henryk's description of "corrugations" (a beautifully descriptive word for this situation), I thought it out, and immediately began piecing together the conneciton. In my mind, these additional fractally branching singularities are due to corrugations as we rotate around any of the singularities.

For the upper primary fixed point, for example, as we rotate clockwise around it, the singularities get closer and closer and the function gets more and more chaotic. On the other hand, if we rotate counter-clockwise, it becomes more and more well-behaved, more and more like the regular slog we would predict if we used only the upper primary fixed point.

This very behavior resembles the slog itself. To the right, it gets more and more fractal, but to the left, it gets smoother and smoother, not unlike the smoothness of the basic exponential in the left half-plane (negative real part).

If we took the slog, with all its smoothness to the left and all its fractal complexity to the right, and "wrapped" that around a logarithmic singularity, we would get the fractal branching in one direction and smoothness in the other direction.

Of course, this is the vaguest of notions so far, a mere hunch, and I need to brush up on complex fourier analysis before I could even begin to tackle this properly.

~ Jay Daniel Fox