You might be tempted, looking at the second image in the previous post, to hope that the white space to the right of the singularity will close off, i.e., that the angle will decrease arbitrarily. In fact, we know that there will always be an angle there, because the bases of the "logarithms" at each singularity are different.
A 180-degree rotation around the singularity at z=2 involves moving \( \frac{\pm\pi}{\ln\ln{2}} \) units in the imaginary direction. Moving 180 degrees around the singularity at z=4, if it were an ordinary logarithmic singularity, would require moving \( \frac{\pm\pi}{-\ln\ln{4}} \) units, about 12% further. If we only moved \( \frac{\pm\pi}{\ln\ln{2}} \) imaginary units around the idealized singularity at z=4, we would only have gone 160.4 degrees. This remaining 19.6 degrees explains, to a first approximation, the white space in the graph (which represents the logarithms of other branches).
To see this, let's zoom in a couple more times, to satisfy ourselves that this angle will indeed appear to remain. Here's a factor of 10 zoom from the previous image:
And another factor of 10:
A 180-degree rotation around the singularity at z=2 involves moving \( \frac{\pm\pi}{\ln\ln{2}} \) units in the imaginary direction. Moving 180 degrees around the singularity at z=4, if it were an ordinary logarithmic singularity, would require moving \( \frac{\pm\pi}{-\ln\ln{4}} \) units, about 12% further. If we only moved \( \frac{\pm\pi}{\ln\ln{2}} \) imaginary units around the idealized singularity at z=4, we would only have gone 160.4 degrees. This remaining 19.6 degrees explains, to a first approximation, the white space in the graph (which represents the logarithms of other branches).
To see this, let's zoom in a couple more times, to satisfy ourselves that this angle will indeed appear to remain. Here's a factor of 10 zoom from the previous image:
And another factor of 10:
~ Jay Daniel Fox
