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.

however i believe in ' conservation of fixed points ' for analytic solutions.

for instance (a^c / b) a^x - c = x has no solution in nonnegative real x and real a,b,c > 1.

thus we linked the superfunction of a function without a real fixpoint to another without a real fixpoint.

a similar thing happens with bases below eta.

we then linked the superfunction of a function with 2 real fixpoints to another with 2 real fixpoints.


as you can see the amount of fixpoints remains constant hence ' conservation of fixed points '.


also intresting might be another example of substitution

exp(x^2 / 2) ^[z] => sqrt( exp^[z] (x^2))

and notice exp(x^2 / 2) = sqrt(exp(x^2)) = sqrt(e)^(x^2)

which leads me to the ' gaussian question '

exp(- x^2 ) ^[z] = ?? however we have a fixpoint there

maybe intresting for statistics and combinatorics ...

regards

tommy1729