Cake is something to turn Tommy into conjectures.

( i dont drink coffee )

Enough jokes.

I conjecture there is a nonzero constant c such that

C f(x) = f(exp(x))

Where f is entire and Clog f is real-analytic.

Some motivation

Take Clog on both sides:

We then get a NEW F :

F(x) + 1 = log(f(exp(x)) / log C.

This is similar to the slog or slog equation.

We know the branch cut difference should be approx. the pseudoperiod P.

So 2pi i / log C = P i assume.

Then C = exp(2 pi i / P).

That Logic is a bit handwaving but it was the motivation.

Related is the idea - not to be confused with question 3 from the OP - ;

There exist nontrivial entire functions E_1,E_2 such that

E_1(slog(x)) = E_2(x).

If the exp only had one primary fixpoint these questions would all be easy.

It inspires to ask these question anyway as if we were unaware of the other fixpoint.

Regards

Tommy1729

( i dont drink coffee )

Enough jokes.

I conjecture there is a nonzero constant c such that

C f(x) = f(exp(x))

Where f is entire and Clog f is real-analytic.

Some motivation

Take Clog on both sides:

We then get a NEW F :

F(x) + 1 = log(f(exp(x)) / log C.

This is similar to the slog or slog equation.

We know the branch cut difference should be approx. the pseudoperiod P.

So 2pi i / log C = P i assume.

Then C = exp(2 pi i / P).

That Logic is a bit handwaving but it was the motivation.

Related is the idea - not to be confused with question 3 from the OP - ;

There exist nontrivial entire functions E_1,E_2 such that

E_1(slog(x)) = E_2(x).

If the exp only had one primary fixpoint these questions would all be easy.

It inspires to ask these question anyway as if we were unaware of the other fixpoint.

Regards

Tommy1729