11/20/2007, 05:15 AM
Apologies for the delay, but it took a while to tweak the code to get useable results.
In the first few posts of this thread, I used a non-linear spacing of the contours to help bring out detail. For the graphs that follow, I've used a linear spacing, meaning contour lines with real parts at constant intervals. In this case, I've chosen 8 contours per real unit.
Internally, I've calculated a "primary branch" of sorts, then calculated the first three branches up and down from there. I then took the logarithms of these six non-primary branches, which gave me the analytic continuation in the primary branch. (By the way, strictly speaking, these aren't branches of the superlogarithm, but correspond to branches of the underlying logarithm for this base. Branches imply branch cuts in a multi-valued relation. As an example, there would be a branch cut in the superlogarithm running between the singularities at z=2 and z=4, which you can't see here because the real contours wrapping around z=2 coincide.)
I then moved these continuations into four non-primary "branches" (first two up and down), and took the logarithm again. I only used four branches this time, rather than 6, because I ran out of memory the first time I tried to run my script. Rather than try to squeeze more points into the same RAM, I opted simply to eliminate points. Anyway, this gave me a second layer of continuation. This process could be repeated indefinitely, had I the CPU time and memory available to do so. (Note: I also took additional logarithms in the primary branch, to fill in some detail.)
First up is a zoomout, so that you can see the next branch up and down from the principal branch:
Next up, I've zoomed in a factor of 2.5, to show a little more detail but still allow you to see the entire branch (top to bottom, not left to right, obviously).
As before, I started with larger images, then shrunk them 50% for bandwidth and screen space. If you'd like to see the originals, let me know and I can email them.
In the first few posts of this thread, I used a non-linear spacing of the contours to help bring out detail. For the graphs that follow, I've used a linear spacing, meaning contour lines with real parts at constant intervals. In this case, I've chosen 8 contours per real unit.
Internally, I've calculated a "primary branch" of sorts, then calculated the first three branches up and down from there. I then took the logarithms of these six non-primary branches, which gave me the analytic continuation in the primary branch. (By the way, strictly speaking, these aren't branches of the superlogarithm, but correspond to branches of the underlying logarithm for this base. Branches imply branch cuts in a multi-valued relation. As an example, there would be a branch cut in the superlogarithm running between the singularities at z=2 and z=4, which you can't see here because the real contours wrapping around z=2 coincide.)
I then moved these continuations into four non-primary "branches" (first two up and down), and took the logarithm again. I only used four branches this time, rather than 6, because I ran out of memory the first time I tried to run my script. Rather than try to squeeze more points into the same RAM, I opted simply to eliminate points. Anyway, this gave me a second layer of continuation. This process could be repeated indefinitely, had I the CPU time and memory available to do so. (Note: I also took additional logarithms in the primary branch, to fill in some detail.)
First up is a zoomout, so that you can see the next branch up and down from the principal branch:
Next up, I've zoomed in a factor of 2.5, to show a little more detail but still allow you to see the entire branch (top to bottom, not left to right, obviously).
As before, I started with larger images, then shrunk them 50% for bandwidth and screen space. If you'd like to see the originals, let me know and I can email them.
~ Jay Daniel Fox
