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 tetration bending uniqueness ? tommy1729 Ultimate Fellow Posts: 1,491 Threads: 355 Joined: Feb 2009 08/28/2010, 11:23 PM (This post was last modified: 08/28/2010, 11:34 PM by tommy1729.) 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 « Next Oldest | Next Newest »

 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

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