using sinh(x) ?

[tommy1729]

also worth mentioning i think :

let the base be a^(1/b) > sqrt(e) so that we can compute

the superfunction of f(x) = a^(1/b)^x with my method.

then consider t(x) = b(x + c)

and its inverse m(x) = x/b - c

m(f(t(x))) = m(a^(x+c)) = (1/b) a^(x+c) - c = (a^c / b) a^x - c

if a , b and c are chosen such that (a^c / b) a^x - c > x

we can compute the superfunction of (a^c / b) a^x - c by computing m(f^[z](t(x))).

regards

tommy1729

if (a^c / b) a^x - c = x we get the intresting case of yet another fixpoint.

associating functions without fixpoint with functions with 1 or more fixpoints is a intresting but complicated idea ...

it raises questions. can we determine the number of superfunctions by that ? can we define uniqueness in some way ?

for instance we associate g , f , k with g no fixpoint , f one fixpoint and k 2 fixpoints.

how many solutions ? 1 ? 2 ? 3 ? 4 ? 5 ? 6 ? oo ?

keep in mind that solution might be equal to eachother.

e.g. expanding f at its fixpoint = expanding k at its second fixpoint.

the key might be to notice that if a function with no fixp has the same superf as the associated with 1 , it may be unique.