06/23/2009, 09:28 PM
(This post was last modified: 06/23/2009, 09:39 PM by Base-Acid Tetration.)
@Kouznetsov: how are the other tetrations relevant to our discussion of uniqueness?
@Kouznetsov: Also I just have a quick question to ask about tetration: What is the mathematical reason that the tetrational approaches the fixed points \( L \) or \( L* \) as you go to \( \pm i \infty \)? I think instead, that \( \lim_{x \rightarrow -\infty} x\pm ci = L\mathrm \) or \( L* \), where c is a nonzero real number, should be the limit that corresponds to the fixed point. (when the imaginary part is zero this doesn't work)
Why is this intuitive reasoning incorrect?:
to get from 1 to a non-real number by iteration of exp, you need complex iteration (real iterations always give real numbers)
But then since \( L,L* \) are repelling with respect to the exponential, you need infinite negative iterations of exp (positive iterations of log). If you infinitely iterate log on a nonreal number, you get closer to L, why isn't this reflected in the tetrational graph? Bo said something like log is not in the initial region anymore, could you clarify that for me? I have attached a schematic diagram to represent my reasoning:
how does the tetrational for base b>exp(1/e) actually behave at large values of real part of non-real arguments?
@Kouznetsov: Also I just have a quick question to ask about tetration: What is the mathematical reason that the tetrational approaches the fixed points \( L \) or \( L* \) as you go to \( \pm i \infty \)? I think instead, that \( \lim_{x \rightarrow -\infty} x\pm ci = L\mathrm \) or \( L* \), where c is a nonzero real number, should be the limit that corresponds to the fixed point. (when the imaginary part is zero this doesn't work)
Why is this intuitive reasoning incorrect?:
to get from 1 to a non-real number by iteration of exp, you need complex iteration (real iterations always give real numbers)
But then since \( L,L* \) are repelling with respect to the exponential, you need infinite negative iterations of exp (positive iterations of log). If you infinitely iterate log on a nonreal number, you get closer to L, why isn't this reflected in the tetrational graph? Bo said something like log is not in the initial region anymore, could you clarify that for me? I have attached a schematic diagram to represent my reasoning:
how does the tetrational for base b>exp(1/e) actually behave at large values of real part of non-real arguments?