Re: " Wanted the r iterated of f(x) = 2*x/(1-x^2) , f^[r](x) "

In article <1128665817.093842.197060@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<alainverghote@xxxxxxxx> wrote:

>Do some of you know a closed form
>for the rth iterate of 2*x/(1-x^2).

.... which equals tan(2 u) if x = tan(u). That is, if f(x)=2x/(1-x^2)
then f(tan(u)) = tan( 2u ), so that f(f(tan(u))) = tan(4u) and
f(f(f(tan(u)))) = tan(8u), etc. In general you can then prove by
induction that f^r ( tan(u) ) = tan( 2^r u ) for positive integers r.

You can write this as f^r( x ) = tan( 2^r arctan(x) ) I suppose but
this is only because, for a given x, the possible values of u all
differ by integer multiples of pi and thus the possible values
of 2^r arctan(x) will also differ by integer multiples of pi, giving
them all the same values in tan() . I say this just to emphasize,
as always must be done, that this analysis is predicated on the
assumption that r is an integer; there is nothing in the defnition
of f alone that tells us what f^r even _means_ when r is
not an integer.

dave





.

Relevant Pages

  • Re: Marginal density of normal+beta random variable
    ... I need the marginal density f. ... Can someone show me how to evaluate this integral so I can obtain a closed form result? ... where W has standard normal distribution. ... This might also be able to do so when a and b are integer multiples of 1/2. ...
    (sci.math)
  • using recursion to create a new list
    ... the integer multiples of that number from 1 to 10, ... (define (multiples-list num) ... ;; Return a list of the square roots of the numbers in a list ... Prev by Date: ...
    (comp.lang.scheme)
  • Re: Sgt Slick.....
    ... >> I happen to think this is a stupid argument. ... >Integer multiples aren't the only multiples that exist my friend. ... ....To be a great NCO, you need three bones: a backbone, a wishbone and a funny bone. ... Prev by Date: ...
    (rec.sport.football.college)
  • Re: Sgt Slick.....
    ... Integer multiples aren't the only multiples that exist my friend. ... Aaron ... Prev by Date: ...
    (rec.sport.football.college)