Thread Rating:
  • 1 Vote(s) - 5 Average
  • 1
  • 2
  • 3
  • 4
  • 5
tetration bending uniqueness ?
#1
uniqueness by the absense of bending points with respect to z in

sexp(slog(z) + r) for positive z and r.

( i use tet for 'my' sexp further )

let

tet(slog(x)) = x

tet(0) = 0 = slog(0)

consider

tet(slog(z) + r)

take derivate with respect to z.

tet'(slog(z) + r) x 1/tet'(slog(z))

take derivate with respect to z.

tet''(slog(z)+r)/tet'(slog(z))^2 - (tet'(slog(z)+r)) * tet''(slog(z))/tet'(slog(z))^3

hence

tet''(slog(z)+r)/tet'(slog(z))^2 = (tet'(slog(z)+r)) * tet''(slog(z))/tet'(slog(z))^3

thus

tet''(slog(z)+r) = tet'(slog(z)+r) * tet''(slog(z))/tet'(slog(z))

hence solve for positive z :

tet''(z+r) = tet'(z+r) * tet''(z)/tet'(z)

make symmetric

tet''(z+r)/tet'(z+r) = tet''(z)/tet'(z)

notice that if a z exists , another one must exist.

thus if for some r , a z exists , there exist oo z solutions.

take integral on both sides ( this step may be a bit dubious ? )

log(tet'(z+r)) + A = log(tet'(z)) + B

hence bending points in

sexp(slog(z) + r) correspond to bending points in sexp(z).

thus

all ( analytic ) sexp(z) with sexp(0) = 0 and positive real to positive real ,

without bending points are identical !!

headscratch ...

regards
Reply


Messages In This Thread
tetration bending uniqueness ? - by tommy1729 - 08/28/2010, 11:23 PM
RE: tetration bending uniqueness ? - by tommy1729 - 08/28/2010, 11:31 PM
RE: tetration bending uniqueness ? - by tommy1729 - 08/29/2010, 06:17 PM
RE: tetration bending uniqueness ? - by tommy1729 - 08/29/2010, 06:38 PM
RE: tetration bending uniqueness ? - by bo198214 - 08/30/2010, 08:56 AM
RE: tetration bending uniqueness ? - by tommy1729 - 08/30/2010, 09:37 AM
RE: tetration bending uniqueness ? - by tommy1729 - 06/06/2011, 10:56 PM
RE: tetration bending uniqueness ? - by bo198214 - 06/07/2011, 07:11 AM
RE: tetration bending uniqueness ? - by bo198214 - 06/07/2011, 07:47 AM
RE: tetration bending uniqueness ? - by mike3 - 06/07/2011, 10:56 AM
RE: tetration bending uniqueness ? - by bo198214 - 06/07/2011, 11:19 AM
RE: tetration bending uniqueness ? - by mike3 - 06/07/2011, 09:10 PM
RE: tetration bending uniqueness ? - by bo198214 - 06/07/2011, 12:27 PM
RE: tetration bending uniqueness ? - by tommy1729 - 06/07/2011, 09:02 PM
RE: tetration bending uniqueness ? - by bo198214 - 06/08/2011, 08:37 PM
RE: tetration bending uniqueness ? - by tommy1729 - 06/08/2011, 12:34 PM
RE: tetration bending uniqueness ? - by tommy1729 - 06/09/2011, 12:26 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  [Exercise] A deal of Uniqueness-critrion:Gamma-functionas iteration Gottfried 6 5,703 03/19/2021, 01:25 PM
Last Post: tommy1729
  A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov 13 22,098 02/17/2017, 05:21 AM
Last Post: JmsNxn
  Uniqueness of half-iterate of exp(x) ? tommy1729 14 28,027 01/09/2017, 02:41 AM
Last Post: Gottfried
  Removing the branch points in the base: a uniqueness condition? fivexthethird 0 3,025 03/19/2016, 10:44 AM
Last Post: fivexthethird
  [2014] Uniqueness of periodic superfunction tommy1729 0 3,466 11/09/2014, 10:20 PM
Last Post: tommy1729
  Real-analytic tetration uniqueness criterion? mike3 25 39,066 06/15/2014, 10:17 PM
Last Post: tommy1729
  exp^[1/2](x) uniqueness from 2sinh ? tommy1729 1 4,228 06/03/2014, 09:58 PM
Last Post: tommy1729
  Uniqueness Criterion for Tetration jaydfox 9 18,184 05/01/2014, 10:21 PM
Last Post: tommy1729
  Uniqueness of Ansus' extended sum superfunction bo198214 4 11,000 10/25/2013, 11:27 PM
Last Post: tommy1729
  A question concerning uniqueness JmsNxn 3 8,859 10/06/2011, 04:32 AM
Last Post: sheldonison



Users browsing this thread: 1 Guest(s)