09/24/2007, 06:16 PM
Note that the a_k I just explained were the ones used in the first post. However, I can see a need for a dual-indexed set, because the slog seems to have singularities at \( 2\pi j i \) offsets from the a_0, so we'd need a second index to distinguish them.
I don't know how long it will take to generate good graphs, but as I've been thinking about the slog, I'm realizing how complex it is. First of all, singularities I once predicted do seem to exist, so there was some small comfort there. They exist on the "logarithmic" branch, accessed for example by going between the singularities at 0.318+1.337i and 0.318+4.946i. Once you've passed between these two singularities, the real line and its offset at 2*pi*i are linear singularities (meaning the entire line is like a "wall").
If you then loop around the singularity again, e.g., between 0.318+1.337i and the real line, then you enter another region, where the real line offset by pi*i is another "wall" singularity. The logarithm of that line is a U-shaped curve (lying on its side), which itself is another singularity. This process continues, looking exactly like the "fractal" graph I posted a few weeks ago. Each of those lines is a 1-dimensional singularity (i.e., not a mere point). I'm not entirely sure what lies on the other side of these singularities, because we can't go "around" them. They extend to infinity at both endpoints. The other side is almost certainly the logarithm from another branch, which would include the singularities with all the strange behavior. Deeper down the rabbit hole we go.
I don't know how long it will take to generate good graphs, but as I've been thinking about the slog, I'm realizing how complex it is. First of all, singularities I once predicted do seem to exist, so there was some small comfort there. They exist on the "logarithmic" branch, accessed for example by going between the singularities at 0.318+1.337i and 0.318+4.946i. Once you've passed between these two singularities, the real line and its offset at 2*pi*i are linear singularities (meaning the entire line is like a "wall").
If you then loop around the singularity again, e.g., between 0.318+1.337i and the real line, then you enter another region, where the real line offset by pi*i is another "wall" singularity. The logarithm of that line is a U-shaped curve (lying on its side), which itself is another singularity. This process continues, looking exactly like the "fractal" graph I posted a few weeks ago. Each of those lines is a 1-dimensional singularity (i.e., not a mere point). I'm not entirely sure what lies on the other side of these singularities, because we can't go "around" them. They extend to infinity at both endpoints. The other side is almost certainly the logarithm from another branch, which would include the singularities with all the strange behavior. Deeper down the rabbit hole we go.
~ Jay Daniel Fox
