10/17/2023, 12:06 PM
Ok here is an idea for uniqueness or at least putting solutions in category.
f(x) is an analytic function on [0,oo] such as analytic semiexp(x) or tet(x), so the idea is general.
More specific
Let f^{n} mean n th derivative of f.
say f(z) satisfies
f(0) = 0
f(1) = 1
f(x+1) = exp(f(x))
f ' (0) = f '(1) = c ( a given c )
then
TR( f(x) ) = integral from 1 to infinity sum_(n>=0) (f^{n}(x))^(-2) dx
LET Min TR( f(x) , c ) = G©
where the minimum is for a fixed c , and f(x) is picked to get a minimum value.
Then we wonder what G© is as a function of c.
And how many analytic solutions f(x) exist with the same G© value.
We can further wonder what value of c makes G© at minimum.
Notice that TR can diverge for non tetration like functions.
Adding a weight might help there.
the details for systematic adding weights needs to be investigated.
regards
tommy1729
f(x) is an analytic function on [0,oo] such as analytic semiexp(x) or tet(x), so the idea is general.
More specific
Let f^{n} mean n th derivative of f.
say f(z) satisfies
f(0) = 0
f(1) = 1
f(x+1) = exp(f(x))
f ' (0) = f '(1) = c ( a given c )
then
TR( f(x) ) = integral from 1 to infinity sum_(n>=0) (f^{n}(x))^(-2) dx
LET Min TR( f(x) , c ) = G©
where the minimum is for a fixed c , and f(x) is picked to get a minimum value.
Then we wonder what G© is as a function of c.
And how many analytic solutions f(x) exist with the same G© value.
We can further wonder what value of c makes G© at minimum.
Notice that TR can diverge for non tetration like functions.
Adding a weight might help there.
the details for systematic adding weights needs to be investigated.
regards
tommy1729
