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Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation
(10/15/2009, 08:17 AM)andydude Wrote: Tetration isn't entire, it's holomorphic over the complex plane cut along the negative reals < -2.
I said PENTation.

(10/15/2009, 05:01 PM)bo198214 Wrote:
(10/13/2009, 09:47 PM)Base-Acid Tetration Wrote: Theorem. There exists no entire pentation pen_b(z), for b>1, such that pen(0) = 1.

Either you have to specify that pen must use the principal branch of tetration (the cut is nothing god given, its just a choice to our human preference; the real function lives on Riemann manifolds), or you have to modify your proof and use the singularities of tetration at integers <=-2 (and show before that indeed every tetration has singularities there).

I DID say:
BAT Wrote:Let Tet_b(z) be the principal branch of tetration.
I fixed the proof to make this clearer.

I also did say that if we didn't restrict ourselves to the principal branch an entire pentation may be possible:
BAT Wrote:Or alternatively we can incorporate parts of non-prinicipal branches of tetration (analytically continued around z = -2) in our construction of a holomorphic pentation?
I meant if it can just avoid -2, pentation can still be entire.

Now the question is, is it morally good to restrict ourselves to the principal branch?

(BTW, what kind of singularities does Tet have at integers<=-2? Are they all logarithmic-type branch points around which the function winds infinitely many times?)


Messages In This Thread
RE: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - by Base-Acid Tetration - 10/15/2009, 09:39 PM

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