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 between addition and multiplication bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 03/11/2011, 10:59 PM (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. « Next Oldest | Next Newest »