Solutions of Doubly and Higher Order Iterated Equations

Michael Fisher once studied the solution of the equation f(f(x))=x2−2. We offer solutions to the general equation f(f(x))=h(x) in the form f(x)=g(ag−1(x)) where a is in general a complex number. This leads to solving duplication formulas for g(x). For the case h(x)=x2−2, the solution is readily found, while the h(x)=x2+2 case is challenging. The solution to these types of equations can be related to differential equations.