Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Regular slog for base sqrt(2) - Using z=2
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 units in the imaginary direction. Moving 180 degrees around the singularity at z=4, if it were an ordinary logarithmic singularity, would require moving units, about 12% further. If we only moved 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

Messages In This Thread
RE: Regular slog for base sqrt(2) - Using z=2 - by jaydfox - 11/15/2007, 11:39 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Inspired by the sqrt tommy1729 0 1,624 02/13/2017, 01:11 AM
Last Post: tommy1729
  Some slog stuff tommy1729 15 13,930 05/14/2015, 09:25 PM
Last Post: tommy1729
  tetration base sqrt(e) tommy1729 2 3,669 02/14/2015, 12:36 AM
Last Post: tommy1729
  Regular iteration using matrix-Jordan-form Gottfried 7 9,320 09/29/2014, 11:39 PM
Last Post: Gottfried
  A limit exercise with Ei and slog. tommy1729 0 2,103 09/09/2014, 08:00 PM
Last Post: tommy1729
  A system of functional equations for slog(x) ? tommy1729 3 4,859 07/28/2014, 09:16 PM
Last Post: tommy1729
  [2014] sqrt boundary tommy1729 0 1,917 06/19/2014, 08:03 PM
Last Post: tommy1729
  slog(superfactorial(x)) = ? tommy1729 3 5,358 06/02/2014, 11:29 PM
Last Post: tommy1729
  [stuck] On the functional equation of the slog : slog(e^z) = slog(z)+1 tommy1729 1 2,709 04/28/2014, 09:23 PM
Last Post: tommy1729
  A simple yet unsolved equation for slog(z) ? tommy1729 0 2,033 04/27/2014, 08:02 PM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)