Schröder's equation

Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function $h$ , find the function $Ψ$ such that

$\forall x\;\;\;\Psi {\big (}h(x){\big )}=s\Psi (x).$

Schröder's equation is an eigenvalue equation for the composition operator $C h$ that sends a function $f$ to $f (h (.))$ .

If $a$ is a fixed point of $h$ , meaning $h (a) = a$ , then either $Ψ(a) = 0$ (or $\infty$ ) or $s = 1$ . Thus, provided that $Ψ(a)$ is finite and $Ψ' (a)$ does not vanish or diverge, the eigenvalue $s$ is given by $s = h' (a)$ .