10/21/2007, 04:11 AM

Okay, to help us better understand the structure of the slog (base e), I think it's important to look at it generically. In other words, let's be ignorant for a moment about the exact values of the slog, and explore instead the branches and their layout relative to each other and relative to fixed points (or singularities, as I'll also typically call them).

To do this, let's start with the following region and call it the "(generic) critical region". The borders of this region are, on the left, the straight line drawn between the two primary fixed points (0.318+1.337i and its conjugate), and on the right, the exponentiation of this line, an arc with center at the origin and endpoints at the fixed points:

Note that this is a fairly generic region, but it suffices for our purposes. No point in the region can be exponentiated without ending up outside the region, and if we take the logarithm of any point in the region, we again end up outside the region. Exponentiate any point on the left boundary, and we'll get a point on the right boundary. Conversely, take the logarithm of any point on the right boundary, and we'll end up with a point on the left boundary. And of course, the fixed points complete the enclosure of this region.

I've color coded this region in a fairly smooth manner, such that green corresponds to the left side of this interval, and blue to the right side. The exact method I used to determine this "smooth" coloring is unimportant, because as I said, this is supposed to be generic.

Now, let's logarithmicize this region. Yes, I made that word up. We end up with the following:

Notice that copies of the new region at the origin appear at 2*pi*i intervals. In fact, though not seen in this relatively small image, there are an infinite number of copies. These correspond to the various branches of the natural logarithm. Exponentiate any one of these regions, or indeed all of them together, and we get back the critical region (or multiple copies of it).

The second thing to notice is that, if we are to maintain some degree of continuity, it only makes sense to make copies of the critical region at 2*pi*i intervals:

And, of course, if these other regions are green-blue (just like our critical region), it only makes sense that a red-green region should appear when we logarithmicize these new regions:

Let's zoom in to try to make out a little more detail:

Still not much to see. Don't worry, it'll start to fit together soon. And by the way, future zooms will not be centered at the origin, for reasons that will be obvious.

You may at this point have doubts that this duplicating of regions is justifiable, but as we go further, you'll see the regions connect smoothly. The "smoothly" part would indicate (qualitatively) that the graph is continuous, and differentiable at least once. I've also given a more mathematical justification elsewhere, but here, we're mainly concerned about getting a quantitative feel for the slog.

To do this, let's start with the following region and call it the "(generic) critical region". The borders of this region are, on the left, the straight line drawn between the two primary fixed points (0.318+1.337i and its conjugate), and on the right, the exponentiation of this line, an arc with center at the origin and endpoints at the fixed points:

Note that this is a fairly generic region, but it suffices for our purposes. No point in the region can be exponentiated without ending up outside the region, and if we take the logarithm of any point in the region, we again end up outside the region. Exponentiate any point on the left boundary, and we'll get a point on the right boundary. Conversely, take the logarithm of any point on the right boundary, and we'll end up with a point on the left boundary. And of course, the fixed points complete the enclosure of this region.

I've color coded this region in a fairly smooth manner, such that green corresponds to the left side of this interval, and blue to the right side. The exact method I used to determine this "smooth" coloring is unimportant, because as I said, this is supposed to be generic.

Now, let's logarithmicize this region. Yes, I made that word up. We end up with the following:

Notice that copies of the new region at the origin appear at 2*pi*i intervals. In fact, though not seen in this relatively small image, there are an infinite number of copies. These correspond to the various branches of the natural logarithm. Exponentiate any one of these regions, or indeed all of them together, and we get back the critical region (or multiple copies of it).

The second thing to notice is that, if we are to maintain some degree of continuity, it only makes sense to make copies of the critical region at 2*pi*i intervals:

And, of course, if these other regions are green-blue (just like our critical region), it only makes sense that a red-green region should appear when we logarithmicize these new regions:

Let's zoom in to try to make out a little more detail:

Still not much to see. Don't worry, it'll start to fit together soon. And by the way, future zooms will not be centered at the origin, for reasons that will be obvious.

You may at this point have doubts that this duplicating of regions is justifiable, but as we go further, you'll see the regions connect smoothly. The "smoothly" part would indicate (qualitatively) that the graph is continuous, and differentiable at least once. I've also given a more mathematical justification elsewhere, but here, we're mainly concerned about getting a quantitative feel for the slog.

~ Jay Daniel Fox