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between addition and multiplication
(03/11/2011, 10:07 PM)martin Wrote:
(03/11/2011, 12:42 PM)bo198214 Wrote: For me it seems important that the curve t |-> a [t] b is smooth (or better analytic) for fixed a and b.
I mean you define it on the interval [1,2], i.e. between addition and multiplication, and then you would continue it to the higher operations t>2 by
a [t+1] (b+1) = a [t] ( a [t+1] b )

And then it would be interesting whether the curve is (infinitely) differentiable at t=2, t=3, etc. Of course it would between the endpoints, i.e. on (2,3) and (3,4), etc.

IIRC, this is exactly what I tried to do, some three years ago or so...
It never really worked out, though some sample graphs for 2 [4] x and 3 [4] x looked quite good.

Hey Martin, we are not talking about the function f(x) = a [4] x but about the function g(x) = a [x] b. But you are right that we expect the same smoothness also for f(x), which is how Andrew Robbins came to his tetration extension.

Messages In This Thread
between addition and multiplication - by lloyd - 03/10/2011, 09:10 PM
RE: between addition and multiplication - by bo198214 - 03/11/2011, 10:59 PM

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