09/25/2009, 12:40 AM

The next step in Kneser's construction is to take the logarithm of the chi function, which gives us a new set of images of the various regions and curves:

At this point, Kneser continues to describe the process, but does not provide further graphs. He is also somewhat vague on details, so I've decided to simply do it the way which seems obvious to me. Note that the L region is periodic, with period equal to c, where c is the (logarithm of the) fixed point.

The obvious choice of how to proceed is to make the periodicity equal to 2*pi*i, so that exponentiating gives a region that overlaps itself perfectly. Here is what the new logarithm looks like:

In order to make it easier to see the detail, I'm going to show a single region, correspond to one full period:

The final step, as should be obvious, is to exponentiate. This yields the following region:

At this point, Kneser continues to describe the process, but does not provide further graphs. He is also somewhat vague on details, so I've decided to simply do it the way which seems obvious to me. Note that the L region is periodic, with period equal to c, where c is the (logarithm of the) fixed point.

The obvious choice of how to proceed is to make the periodicity equal to 2*pi*i, so that exponentiating gives a region that overlaps itself perfectly. Here is what the new logarithm looks like:

In order to make it easier to see the detail, I'm going to show a single region, correspond to one full period:

The final step, as should be obvious, is to exponentiate. This yields the following region:

~ Jay Daniel Fox