11/15/2007, 11:06 AM

Before I start showing branches, I wanted to show you zoom-ins of the singularity at z=4. I believe the term for it is essentialy singularity, as there is a line of singularities approaching it from the right, a countably infinite number of singularities that have z=4 as their limit point. Thus, we cannot remove this singularity in the traditional sense. The singularity at 2 is removable, in the sense that a Taylor series developed arbitrarily close to z=2 would have a radius of convergence limited by the singularity at z=4. I'm not sure if it's valid to develop a power series at z=2 after removing the singularity, however.

Anyway, here's the first zoomin, showing both singularities, to help you appreciate the difference between them:

Notice that the red line in the middle is fairly straight. Somewhere in that vicinity, the curvature of the contours at the real points switches from left-curving to right-curving. This is a point of inflection for the rslog on the real interval between z=2 and z=4. I'm not sure what if any significance this has, but it is similar in principle to the point of inflection on the critical interval for base e, and it's worth further exploration at some point in the future when one of us can get around to it.

Next, we'll zoom in properly on the singularity at z=4:

Anyway, here's the first zoomin, showing both singularities, to help you appreciate the difference between them:

Notice that the red line in the middle is fairly straight. Somewhere in that vicinity, the curvature of the contours at the real points switches from left-curving to right-curving. This is a point of inflection for the rslog on the real interval between z=2 and z=4. I'm not sure what if any significance this has, but it is similar in principle to the point of inflection on the critical interval for base e, and it's worth further exploration at some point in the future when one of us can get around to it.

Next, we'll zoom in properly on the singularity at z=4:

~ Jay Daniel Fox