Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
fake id(x) for better 2sinh method.
For improving the 2sinh method so that it works for all bases > eta and other improvements I consider :

2sinh(fake id(x))

fake id(x) behaves like e/2 x for small x, like x for large real and like 0 for large imaginary part.

SO what is the best fake id(x) ? Remember we only want a single fixpoint (2sinh(fake id(x))=x) for any base.

maybe id(x) = exp(fakelog(x)) ...



There is a problem with this particular type of fake function.
Although it may be fixed as stated in post 1 it seems a big problem.

Perhaps the biggest problem of " fake function theory ".

id(x) as desired in post nr 1 : " fake id(x) behaves like e/2 x for small x, like x for large real and like 0 for large imaginary part "

is in contradiction with the well known Lindelöf theorem.

It seems the natural fix is : " fake id(z) behaves like e/2 z for small z and like z for large z "


Hmm this new id(z) also seems in trouble because

id(z) -z goes to 0 for large z ... but stays bounded for small z : so id(z) - z is not entire.

So maybe use a function that grows like exp(x) near the real line and grows like z otherwise ?

Notice if id(z) is not entire , then neither is id(exp(z)) or even id(fakeexp(z)).


Seems really an issue.


Wait a sec , it seems carlson's theorem applies here !

id(z) must grow fast enough otherwise id(z) - z is "close" to 0 for integer z , hence to avoid id(z) being flat it seems id(z) must grow fast enough in the imaginary direction.

The problem is " close " is not the exact condition of the theorem
(exactly 0 is).

To make ln(id(z)/z) an entire function this also suggests id(z) is of exp type.

Hence I write id(z) = z exp(t(z)) where t(z) is a taylor series.


and exp(t(x)) for real x is a fake id(1) function.

Thus t(x) is a fake id(0) function.

However saying t(z) = id(z) - z brings us back to the Original problem ...


This is getting stranger by the sec.

Maybe t(z) should be a fake id(0) for reals and a fake exp^[1/2](z) in the upper half plane ( in terms of absolute value ).
But that brings us back again to carlson so I guess its better to have

f(z) = fake id(0) around the real axis.
f(z) = fake exp(z) near the imag axis.

I finally see less objections.
But what is f(z) ?


(06/04/2014, 11:10 PM)tommy1729 Wrote: Now id(z)/z IS NEVER ALLOWED TO BE EQUAL TO 1.

and exp(t(x)) for real x is a fake id(1) function.

This is still a big objection.

so exp(t(z)) is not allowed to be 0 nor 1.
This violates the little picard theorem.


Seems all theorems of complex analysis are against me ! Smile


A good id(z) has been found !

id(z)/z = 1 sometimes but that is no big problem.

id(z) = z * fake(1) on the real line
id(z) = z * fake(exp(exp(i z))^O(z)) on the imag line.

id(z) is of the type z * f(z i) where f(z i) is similar too

Yes the same function that occured in the fake exp^[1/2] thread.

It seems " fake function theory " is developping fast. Smile




Possibly Related Threads...
Thread Author Replies Views Last Post
  Doubts on the domains of Nixon's method. MphLee 1 61 Yesterday, 10:43 PM
Last Post: JmsNxn
  My interpolation method [2020] tommy1729 1 1,653 02/20/2020, 08:40 PM
Last Post: tommy1729
  Kneser method question tommy1729 9 6,553 02/11/2020, 01:26 AM
Last Post: sheldonison
  Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 1,407 09/09/2019, 10:55 PM
Last Post: tommy1729
  tommy's simple solution ln^[n](2sinh^[n+x](z)) tommy1729 1 4,058 01/17/2017, 07:21 AM
Last Post: sheldonison
  2 fixpoints , 1 period --> method of iteration series tommy1729 0 2,647 12/21/2016, 01:27 PM
Last Post: tommy1729
  Tommy's matrix method for superlogarithm. tommy1729 0 2,640 05/07/2016, 12:28 PM
Last Post: tommy1729
  2sinh^[r](z) = 0 ?? tommy1729 0 2,093 02/23/2016, 11:13 PM
Last Post: tommy1729
  [split] Understanding Kneser Riemann method andydude 7 12,468 01/13/2016, 10:58 PM
Last Post: sheldonison
  Kouznetsov-Tommy-Cauchy method tommy1729 0 2,954 02/18/2015, 07:05 PM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)