03/15/2011, 11:21 PM

Tommy, the relevance to tetration is that tetration is an attempt to extend the sequence a{1}b, a{2}b, a{3}b... (where these are respectively addition, multiplication, exponentiation) to a{4}b (tetration) -- but another way to extend the sequence is to find the functions where phi (which I use for the value value in curly brackets) is inbetween the integer values we already know. Your guesses that x {0.5} x = x!, and x{y}x = x are wrong and do not follow from the definitions. It's true there is no closed form for these intermediate operations as I defined them yet, and it's true that I didn't know Gauss had invented the arithmetic-geometric mean, if that matters to you. But look at the graphics I posted and tell me you don't think the numerical solutions using these weighted functions look good as interpolated curves.

Finding closed forms will be very difficult, as already the closed form for Gauss's mean involves some tough steps.

Finding closed forms will be very difficult, as already the closed form for Gauss's mean involves some tough steps.

(03/15/2011, 10:22 PM)tommy1729 Wrote: if i understand this confusing thread well

x {0.5} x = x !

in fact x {y} x = x

at least following from

( quote )

In other words, for a {0.25} b, with a<b,

m1 of a and b = a + (b-a)*0.25 (0.25 of the way between a and b, judged arithmetically)

m2 of a and b = a * (b/a)^0.25 (0.25 of the way between a and b, judged geometrically)

Now plug m1 and m2 into a and b, and iterate until you get something stable (i.e. m1 = m2 to whatever degree of precision you need)

( end quote )

unless you guys started using different ideas since the time of the quote.

furthermore i think this has nothing to do with * tetration * and see it more like an idea inspired by Gauss Aritmetic-Geometric Mean.

If this was not my favorite forum and due to the lack of good math forums in general imho , i might not have read it in the first place ( OP was already unaware of Gauss AGM Mean ) although it might get intresting soon ...

i would be more carefull to associate this immediately with a new * slog *.

i think JmsNxn is a bit overenthousiastic.

maybe its me , but the only intresting thing i can see at the moment is :

does this ' new ' mean have a closed form similar to gauss his result ?

( i assume its analytic ?? for complex z_i : z1 {z2} z3 ? )

furthermore , i have no idea why JmsNxn thinks x {1.5} 2 = x {0.5} x

???

tommy1729