Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Iterated tangent

  1. Oct 5, 2007 #1


    User Avatar
    Science Advisor
    Homework Helper

    Let [itex]\tan_1x=\tan x[/itex] and [itex]\tan_{k+1}x=\tan\tan_kx[/itex].

    It's fairly clear that the sequence [itex](\lfloor\tan_n1\rfloor)[/itex] = http://www.research.att.com/~njas/sequences/A000319 [Broken] is chaotic, in the sense that it can diverge from [itex](\lfloor\tan_n(1+\varepsilon)\rfloor)[/itex] even for small [itex]\varepsilon[/itex].

    1. Any thoughts on how to calculate members of this sequence efficiently? The loss of precision at each step makes this very difficult to calculate; also, the tangent is difficult to compute compared to more elementary functions. For comparison's sake, 4000 decimal digits of precision are needed to calculate a(2,000,000), which takes perhaps a week in naive implementation on Pari.
    2. Does any proof idea spring to mind regarding the idea that for all integers k, a(n) = k for some n? This seems natural in light of the chaotic nature of the problem... but I have nothing in mind.
    3. Can anyone thing of a good explanation for the 'probability' that a random member of the sequence will be equal to a given number k? Assuming some kind of equidistribution I thought it was reasonable to consider
    but this seems to vastly overestimate the chances for large k. To wit, it would predict that about 91 out of every million randomly selected elements would be in the range 190-200, but in the first 3 million or so elements I haven't found one.
    Last edited by a moderator: May 3, 2017
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted