Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
levy ecalle koenigs ?
#5
(07/09/2010, 12:27 PM)tommy1729 Wrote: why is the radius zero again ?

cant find your paper about parabolic iteration ...

Actually I dont know exactly the proof, but in most cases the convergence radius is 0 for parabolic iteration, particularly for the iteration of e^x-1 which is in turn equivalent to iteration of e^(x/e).

For parabolic Abel function there is a very old formula by Lévy (2.20 in the overview paper), which is:


which however is not usable for numeric calculation as it is too slow.
A formula given by Ecalle (2.22 in the overview paper) is much more usable and works for both cases hyperbolic and parabolic. It is

where is a sum of some negative powers (none in the hyperbolic case) and a logarithm for example for e^x-1 we get:
.

Another formula (2.29 in the overview paper) that kinda combines hyperbolic and parabolic is:
, ,

where is the derivative at the fixed point 0, which is 1 in the parabolic case and you take the limit of lambda->1. I am in a hurry a bit. So perhaps more detailed later.
Reply


Messages In This Thread
levy ecalle koenigs ? - by tommy1729 - 07/08/2010, 11:31 PM
RE: levy ecalle koenigs ? - by bo198214 - 07/09/2010, 05:46 AM
RE: levy ecalle koenigs ? - by tommy1729 - 07/09/2010, 12:27 PM
RE: levy ecalle koenigs ? - by bo198214 - 07/10/2010, 05:17 AM
RE: levy ecalle koenigs ? - by sheldonison - 07/09/2010, 11:48 AM
RE: levy ecalle koenigs ? - by tommy1729 - 07/21/2010, 10:41 PM
RE: levy ecalle koenigs ? - by bo198214 - 07/24/2010, 02:59 AM



Users browsing this thread: 1 Guest(s)