  Using infinite compositions, we solve the general equations $P(\lambda w) =
p(w)f(P(w))$ for holomorphic functions $p$ and $f$. We describe the situations
in which this equation is palpable; and their effectiveness at describing
dynamical properties of the orbit $f^{\circ n}(z)$. We similarly make a change
of variables to study a generalized form of the Abel equation, $F(s+1) =
u(s)f(F(s))$. This paper is intended as a more in depth examination of work
done previously in our last paper--The Limits of a Family; Of Asymptotic
Solutions to the Tetration Equation.
