Riemann surface equation RB ( f(z) ) = f( p(z) )

[tommy1729]

Let f(z) be an analytic function with a riemann surface that has branches.

Let RB be shorthand for going from one specific Riemann Branch to another specific one.

Then the equation that holds locally or globally :

RB ( f(z) ) = f( p(z) )

with p(z) a degree 1 or 2 polynomial.

fascinates me.

For instance f(2z+1) is another branch of f(z).
That is fascinating.

But how to solve such a thing ?

in particular

RB ( f(z) ) = f(z + C)

This relates to tetration and dynamics.

Not sure if its in the books.

Btw do not confuse with a simple invariant :

p (f(z)) is another branch of f(z).

which is different !!


regards

tommy1729