Thread Rating:
  • 1 Vote(s) - 5 Average
  • 1
  • 2
  • 3
  • 4
  • 5
2 fixpoints , 1 period --> method of iteration series
#1
Some credit Goes to Gottfried.

Assume a strictly increasing analytic function f(x) with fixpoints A and B such that

B > A > 0 and

F(x) > 0 for x > 0

And also the Period of the first fixpoint = the Period of the second fixpoint.

( notice this is possible because Period Q = Period - Q !!! Example f ' (A) = 1/ f ' (B) )

Now we construct a function that can be used as a kind of slog type , including if necc BRANCHES for the FUNCTIONAL inverse to make iterations of f.

Call the slog type function T

Then we have for a suitable C whenever f is not growing too fast :
( C depends on the growth of f and the derivate at the fixpoints )

T(x) = ... Rf^[-2] c^2 + Rf^[-1] C + R ( x ) + Rf(x)/C + R f^[2](x)/c^2 + ...

Where Rf^[y] means R ( f^[y] (x) ) and R(x) = (x-a)(x-b).

Now we split Up t(x) into t1(x) + t2(x) in the obvious way ( separate positive and negative iterates )

Such that we arrive at

T(x) = t1(x) + t2(x)
T(f(x)) = t1(x) C + t2(x) / C.

So finally ( again for inverse use branches )

We get the slog type equation

T( f(x)^[y] ) = t1(x) C^y + t2(x) C^{-y}

As desired.

Since the periods agree and we can use analytic continuation we have a solution that is analytic in the interval [A, +oo] , assuming that T is analytic in a nonzero radius.

So we have a solution agreeing on 2 fixpoints !!

Regards

Tommy1729
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Arguments for the beta method not being Kneser's method JmsNxn 49 2,807 10/13/2021, 05:01 AM
Last Post: JmsNxn
  The Generalized Gaussian Method (GGM) tommy1729 0 94 09/25/2021, 12:24 PM
Last Post: tommy1729
  tommy's singularity theorem and connection to kneser and gaussian method tommy1729 2 218 09/20/2021, 04:29 AM
Last Post: JmsNxn
  Why the beta-method is non-zero in the upper half plane JmsNxn 0 202 09/01/2021, 01:57 AM
Last Post: JmsNxn
  Tommy's Gaussian method. tommy1729 20 2,446 08/19/2021, 09:40 PM
Last Post: tommy1729
  Improved infinite composition method tommy1729 5 864 07/10/2021, 04:07 AM
Last Post: JmsNxn
  A different approach to the base-change method JmsNxn 0 595 03/17/2021, 11:15 PM
Last Post: JmsNxn
  A support for Andy's (P.Walker's) slog-matrix-method Gottfried 4 4,533 03/08/2021, 07:13 PM
Last Post: JmsNxn
  Doubts on the domains of Nixon's method. MphLee 1 838 03/02/2021, 10:43 PM
Last Post: JmsNxn
  Perhaps a new series for log^0.5(x) Gottfried 3 4,105 03/21/2020, 08:28 AM
Last Post: Daniel



Users browsing this thread: 1 Guest(s)