09/24/2007, 05:37 AM
I'll create a separate thread for discussion of the rational coefficients I found above. While unexpected and intriguing, I have no idea if they have anything to do with tetration. More than likely it's just a fascinating property of the set of fixed points of base e.
As for the slog: this function is more and more intriguing, the more I study it.
If you look at the standard logarithm, one thing stands out to me. It's the "branches". I'm not sure "branch" is the best description.
This isn't new, but it's worth repeating, as the analogy with the slog breaks down in subtle ways. For the logarithm, the derivative at any point is pretty much fixed, regardless of the branch. If you integrate the derivative on a closed loop, you get back to the original point, so long as you don't go around the singularity. If you do go around the singularity, then when you get back to your starting point, you're either up or down a branch (unless you went around multiple times, in which case you might be up or down multiple branches).
However, regardless of which branch you're on, everything looks the same, other than the constant difference in "height" ("height" being a complex offset, a multiple of \( 2\pi i \) for base e).
Well, the slog would appear to work differently. First, we must define the slog very loosely. Essentially, define it by saying that slog(ln(z)) = slog(z)-1 on some branch.
For example, slog(0.25+1.25i) might be (pulling a number out of thin air) -0.5+2i. If we exponentiate 0.25+1.25i, we get about 0.4+1.2i, and the slog of that should be about 0.5+2i.
Now, if we take the logarithm of 0.25+1.25i, we'll get about 0.25+1.37i, the slog of which should be about -1.5+2i.
Now let's look at the point 1.325+1.307i. The slog of this point might be something like 0+3.1i. However, this point happens to be approximately the fourth iterated logarithm of 0.25+1.25i. Therefore, we would have expected an slog of -4.5+2i.
What happened? Well, we switched branches, because the fourth iterated logarithm took us up and around the singularity. If we have a power series constructed at the origin, it will only give us values that can be reached without going around the singularity.
So far, this isn't terribly interesting. We get the same behavior with the natural logarithm, when we rotate points about the origin. However, where the slog really stands out, is that the derivative is not the same after going around the singularity. If we start at 0.25+1.25 i, then loop around the singularity, we arrive in a totally new landscape.
This is easy to see if we remember the fractal nature of the logarithms of the unit interval (0, 1). The second logarithm is a straight line from \( \infty+\pi i \) to \( -\infty+\pi i \), using the principal branch. Notice that this straight line implies that, excepting a complex constant term, the power series of the slog constructed at a point with imaginary part \( \pi i \) would have real coefficients. Furthermore, this implies that the upper and lower halves (relative to the point) are mirror images of each other (complex conjugates). In other words, it's symmetric about the line y=a+pi*i. But we can use other branches of the natural logarithm to show that it is symmetric about all lines a+pi*(2k+1)*i.
And of course, this symmetry necessarily implies symmetry about all lines a+pi*(2k)*i as well, because the real line reflected about an line with constant imaginary part will be another line with constant imaginary part.
Now, all this is assuming that we're looking at things from the point of view of the branch which includes the origin. If, from the origin, we go between two singularities (there will be singularities at the primary fixed points, and due to the symmetry, there must be singularities at 2k*pi*i offsets from the primary singularities), we end up in another world, so to speak.
If we go between the singularities on either side of the line a+pi*(2k)*i, we end up in the exponential world. Here we get the iterated exponentials of the circular critical region about the origin.
If, from the origin, we go up or down and then between the singularities on either side of a+pi*(2k+1)*i, we end up in the logarithmic world. Here we find the fractal world I described in another topic.
Curiously enough, each world must in some way affect the other, because the power series at any point, while unable to calculate points outside the radius of convergence, must nonetheless "mesh" with a region calculated from the power series based off a neighboring point. Each world, therefore, affects the other, as its derivatives will influence the behavior towards and around the singularity, like sound waves refracting around a corner. Imagine sound waves refracting around the pillar of a spiral staircase, with the steps of the stairs preventing direct interaction from above or below.
As for the slog: this function is more and more intriguing, the more I study it.
If you look at the standard logarithm, one thing stands out to me. It's the "branches". I'm not sure "branch" is the best description.
This isn't new, but it's worth repeating, as the analogy with the slog breaks down in subtle ways. For the logarithm, the derivative at any point is pretty much fixed, regardless of the branch. If you integrate the derivative on a closed loop, you get back to the original point, so long as you don't go around the singularity. If you do go around the singularity, then when you get back to your starting point, you're either up or down a branch (unless you went around multiple times, in which case you might be up or down multiple branches).
However, regardless of which branch you're on, everything looks the same, other than the constant difference in "height" ("height" being a complex offset, a multiple of \( 2\pi i \) for base e).
Well, the slog would appear to work differently. First, we must define the slog very loosely. Essentially, define it by saying that slog(ln(z)) = slog(z)-1 on some branch.
For example, slog(0.25+1.25i) might be (pulling a number out of thin air) -0.5+2i. If we exponentiate 0.25+1.25i, we get about 0.4+1.2i, and the slog of that should be about 0.5+2i.
Now, if we take the logarithm of 0.25+1.25i, we'll get about 0.25+1.37i, the slog of which should be about -1.5+2i.
Now let's look at the point 1.325+1.307i. The slog of this point might be something like 0+3.1i. However, this point happens to be approximately the fourth iterated logarithm of 0.25+1.25i. Therefore, we would have expected an slog of -4.5+2i.
What happened? Well, we switched branches, because the fourth iterated logarithm took us up and around the singularity. If we have a power series constructed at the origin, it will only give us values that can be reached without going around the singularity.
So far, this isn't terribly interesting. We get the same behavior with the natural logarithm, when we rotate points about the origin. However, where the slog really stands out, is that the derivative is not the same after going around the singularity. If we start at 0.25+1.25 i, then loop around the singularity, we arrive in a totally new landscape.
This is easy to see if we remember the fractal nature of the logarithms of the unit interval (0, 1). The second logarithm is a straight line from \( \infty+\pi i \) to \( -\infty+\pi i \), using the principal branch. Notice that this straight line implies that, excepting a complex constant term, the power series of the slog constructed at a point with imaginary part \( \pi i \) would have real coefficients. Furthermore, this implies that the upper and lower halves (relative to the point) are mirror images of each other (complex conjugates). In other words, it's symmetric about the line y=a+pi*i. But we can use other branches of the natural logarithm to show that it is symmetric about all lines a+pi*(2k+1)*i.
And of course, this symmetry necessarily implies symmetry about all lines a+pi*(2k)*i as well, because the real line reflected about an line with constant imaginary part will be another line with constant imaginary part.
Now, all this is assuming that we're looking at things from the point of view of the branch which includes the origin. If, from the origin, we go between two singularities (there will be singularities at the primary fixed points, and due to the symmetry, there must be singularities at 2k*pi*i offsets from the primary singularities), we end up in another world, so to speak.
If we go between the singularities on either side of the line a+pi*(2k)*i, we end up in the exponential world. Here we get the iterated exponentials of the circular critical region about the origin.
If, from the origin, we go up or down and then between the singularities on either side of a+pi*(2k+1)*i, we end up in the logarithmic world. Here we find the fractal world I described in another topic.
Curiously enough, each world must in some way affect the other, because the power series at any point, while unable to calculate points outside the radius of convergence, must nonetheless "mesh" with a region calculated from the power series based off a neighboring point. Each world, therefore, affects the other, as its derivatives will influence the behavior towards and around the singularity, like sound waves refracting around a corner. Imagine sound waves refracting around the pillar of a spiral staircase, with the steps of the stairs preventing direct interaction from above or below.
~ Jay Daniel Fox
