Given holomorphic functions satisfying the functional equation φ = σ ◦ φ ◦ τ where τ has an attracting
fixed point paired with a repelling fixed point of σ, we prove φ can be expressed as a composition expansion
limn→∞ σ ◦ n ◦ ψ ◦ τ ◦ n where ψ approximates φ in some sense. With certain restrictions, φ is the unique function
satisfying the functional equation. Conversely, given a functional equation of the specified form, we construct
a function which satisfies it. The idea behind the proof is to view the transformation f → σ ◦ f ◦ τ as
a contraction mapping on a particular space of holomorphic functions. As a basic example, the functional
equation cos z = 2 cos2 (z/2) − 1 generates a composition expansion for cos z.