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 Real and complex behaviour of the base change function (was: The "cheta" function) bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 08/18/2009, 08:47 AM (This post was last modified: 08/18/2009, 01:33 PM by bo198214.) Just want to clarify the relation between the decremented exponential $\operatorname{dexp}(x):=e^x-1$ and $\exp_\eta$ with respect to base change by giving the formula. We know that they are affine conjugates: $\operatorname{dexp} = \tau^{-1}\circ \exp_\eta \circ \tau$ where $\tau(x)=(x+1)e$. We also have $\log(\log(\exp_\eta(\exp_\eta(x))))=\log(e^{-1}\exp_\eta(x))=-1+e^{-1}x=\tau^{-1}(x)$ thatswhy: $\log^{[n]}\circ\exp_\eta^{[n]}=\log^{[n-2]}\circ\tau^{-1}\circ \exp_\eta^{[n-2]}\circ \tau\circ \tau^{-1}=\log^{[n-2]}\circ(\tau^{-1}\circ \exp_\eta\circ \tau)^{[n-2]}\circ \tau^{-1}$ So this is the relation $\fbox{\log^{[n]}\circ\exp_\eta^{[n]} =\log^{[n-2]}\circ\operatorname{dexp}^{[n-2]}\circ \tau^{-1}}$ where $\tau^{-1}(x)=x/e-1$. Update: This formula can be generalized to a base change from $b=a^{1/a}$ to $a$, i.e. from a base to one of its fixed points: $\fbox{\log_a^{[n]}\circ\exp_b^{[n]} =\log_a^{[n-2]}\circ\operatorname{dexp}_a^{[n-2]}\circ \tau^{-1}}$ where $\tau^{-1}(x)=x/a-1$ and $\operatorname{dexp}_a(x)=a^x-1$. « Next Oldest | Next Newest »

 Messages In This Thread Real and complex behaviour of the base change function (was: The "cheta" function) - by bo198214 - 08/12/2009, 08:59 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function) - by jaydfox - 08/15/2009, 12:54 AM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 05:00 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/15/2009, 05:36 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 06:40 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 07:13 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/15/2009, 09:44 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 10:40 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 10:46 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 11:02 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 11:20 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/16/2009, 11:15 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/16/2009, 11:38 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/17/2009, 08:50 AM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/17/2009, 12:07 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by sheldonison - 08/17/2009, 04:01 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/17/2009, 04:30 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/17/2009, 05:26 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by sheldonison - 08/18/2009, 04:37 AM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/17/2009, 05:47 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function) - by bo198214 - 08/17/2009, 02:40 PM base change with decremented exponential - by bo198214 - 08/18/2009, 08:47 AM singularities of base change eta -> e - by bo198214 - 08/18/2009, 06:51 PM RE: singularities of base change eta -> e - by bo198214 - 08/20/2009, 10:28 AM RE: Does the limit converge in the complex plane? - by sheldonison - 08/13/2009, 12:49 AM RE: Does the limit converge in the complex plane? - by bo198214 - 08/13/2009, 07:17 AM RE: Does the limit converge in the complex plane? - by sheldonison - 08/13/2011, 10:32 AM RE: Does the limit converge in the complex plane? - by bo198214 - 08/13/2011, 06:33 PM RE: Does the limit converge in the complex plane? - by sheldonison - 08/13/2009, 06:48 PM

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