01/04/2010, 06:08 AM
(This post was last modified: 01/04/2010, 06:09 AM by sheldonison.)
(01/04/2010, 03:51 AM)mike3 Wrote: Even thought it seems the the tet and cheta functions do this "ringing", how do you know two tet functions (to different bases) also do it? Remember how different they are on the complex plane.All of the bases show the same behavior, though more graphs might help to demonstrate it.
The graph shows cheta lined up with five different bases, all exactly lined up at x=5.884, so that cheta(x) = sexp_b(x+k_b) = sexp_c(x+k_c) for all five bases. Since for increasing x, slog_a(sexp_b(x+1)) converges to slog_a(sexp_b(x))+1, the five super exponentials also agree again, at around x=6.884, 7884, 8.884, 9.884, etc. But, they disagree in between, and they disagree by different amounts with each base having a different magnitude of ringing in the slog (inverse sexp) domain, so they can't agree with each other either. In the slog domain, the magnitude of the ringing for cheta against base e is +/- 0.04%, but the magnitude for cheta against base 1.464 is only +/- 0.0015%, so sexp_e can't agree with sexp_1.464 in between either. Note, that as would be expeced, the magnitude of the ringing for slog_e(cheta) is the same as for inv_cheta(sexp_e).
Also, all the bases are different in the complex plane too, with an infinity of singularities, as you repeatedly iterate logarithms of one base against the sexp of another base, though the only case that was shown was cheta and sexp_e. Also, ringing was shown for the four super functions of exp(sqrt(2)), see the bummer thread., so perhaps this shouldn't be as disturbing as it initially seems.
- Shel
