Triggered by the interesting finding of Andrew I open this thread for the further investigation of elementary superfunctions, i.e. functions
that are expressible with elementary functions and operations such that
=f(F(x)))
for a given elementary function
.
Our first example is:
with a superfunction
.
Now the
has the same property
as the
. Hence
is another superfunction of
.
Indeed
exists and is differentiable at
.
But it does not exist at the other fixed point
, because
is not defined.
Edit: both are regular super-functions at fixed point 1.
.
So if we are at polynomials
, we can also give an elementary superfunction for
, i.e.
.
Because
.
Edit: these are the regular super-exponentials at 1.
.
Generally for Chebyshev polynomials, these are the polynomials
such that
- for example above we used
-, we know already two elementary superfunctions of
, these are
and
.
for a given elementary function
Our first example is:
Now the
Indeed
But it does not exist at the other fixed point
Edit: both are regular super-functions at fixed point 1.
So if we are at polynomials
Because
Edit: these are the regular super-exponentials at 1.
Generally for Chebyshev polynomials, these are the polynomials