Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Theorem on tetration.
#1
Hey everyone. I thought I'd post this theorem, perhaps someone has some uses for it.

Theorem:

A.) If is holomorphic for for some and for and .

B.) for some and we have

Then, for we have

Proof:

Well this is rather easy:



Which follows by cauchy's residue formula and the bounds of F (the gamma function along with x small enough pulls the arc next to our line integral to zero at infinity). For those who don't see,



where the right term is entire in z and only contribute asymptotics, observe stirlings asymptotic formula



Therefore this holds.


Now observe that by a similar argument:




And of course, by another similar argument:



Therefore since the kernel of this integral transform is zero (its a modified fourier transform). On the line we have . Therefore since both functions are analytic we get the desired.



I'm wondering, does anyone see any uses for this?

I know with some formal manipulation we can say that, if and and is holo and is invertible which satisfies the bounds above. Then
Reply


Messages In This Thread
Theorem on tetration. - by JmsNxn - 06/09/2014, 02:47 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Theorem in fractional calculus needed for hyperoperators JmsNxn 5 6,185 07/07/2014, 06:47 PM
Last Post: MphLee
  [2014] tommy's theorem sexp ' (z) =/= 0 ? tommy1729 1 2,671 06/17/2014, 01:25 PM
Last Post: sheldonison
  Vincent's theorem and sin(sexp) ? tommy1729 0 1,643 03/22/2014, 11:46 PM
Last Post: tommy1729
  number theory EPNT : extended prime number theorem tommy1729 0 1,884 08/23/2012, 02:44 PM
Last Post: tommy1729
  Carlson's theorem and tetration mike3 8 9,113 08/22/2010, 05:12 AM
Last Post: bo198214
  Weak theorem about an almost identical function: PROOF Kouznetsov 3 5,324 11/14/2008, 01:11 PM
Last Post: Kouznetsov



Users browsing this thread: 1 Guest(s)