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+--- Thread: Grand Unity Conjecture (/showthread.php?tid=341)
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RE: Grand Unity Conjecture - bo198214 - 09/07/2009
(09/07/2009, 06:16 PM)Ansus Wrote: Is it already proven that holomorphic tetration is unique?
You mean holomorphic = holomorphic on \( \mathbb{C}\setminus (-\infty,-2] \)?
There is even an entire tetration (regular sexp at a complex fixed point) which though is not real on the real axis.
Dmitrii's approach via lemma about almost identical functions (also somewhere on the forum) has gaps yet.
RE: Grand Unity Conjecture - Base-Acid Tetration - 09/08/2009
Quote:There is even an entire tetration (regular sexp at a complex fixed point) which though is not real on the real axis.does the entire tetration cheat by using different branches of logarithm? like, for the sake of argument, tet(0)=1, but tet(-1)= \( 2k \pi i \) for some integer k
whatevs.
why don't we adopt the entire tetration?
To answer my own question:
It just feels so perverse having nonreal values at INTEGERs but, again, whatevs.
RE: Grand Unity Conjecture - bo198214 - 09/08/2009
(09/08/2009, 05:57 AM)Tetratophile Wrote: does the entire tetration cheat by using different branches of logarithm? like, for the sake of argument, tet(0)=1, but tet(-1)= \( 2k \pi i \) for some integer k
No, haha, it doesnt cheat. An entire function takes on every value infinitely often. Particularly the value 0, say sexp0(z_0)=0. Then you can just shift the function letting sexp(z)=sexp0(z0+1+z), then sexp(-1)=0. And by the equation sexp(z+1)=exp(sexp(z)) you get real values on all integers > -2.
Quote:why don't we adopt the entire tetration?
Actually Kneser constructed from that entire tetration a real analytic tetration (for details see my post with Kneser in the title).
RE: Grand Unity Conjecture - Base-Acid Tetration - 09/08/2009
Quote:No, haha, it doesnt cheat. An entire function takes on every value infinitely often. Particularly the value 0, say sexp0(z_0)=0. Then you can just shift the function letting sexp(z)=sexp0(z0+1+z), then sexp(-1)=0. And by the equation sexp(z+1)=exp(sexp(z)) you get real values on all integers > -2.
i used the word entire to destinguish that tetration from the real ordinary tetration.
i thought entire meant holomorphic everywhere. if the entire tetration is with real values at integers, it can't be differentiable at z=-2
RE: Grand Unity Conjecture - bo198214 - 09/08/2009
Quote:if the entire tetration is with real values at integers, it can't be differentiable at z=-2
Oh, you are right with that. My mistake. An entire takes on every value at most *except* one, and this seems to be 0 here.
Yes then you are right, you can choose it such that sexp(0)=1, but it then would not satisfy sexp(-1)=0, but that sexp(-1)=2*pi*k for some nonzero k.