Functional equation

Summary

A functional equation is an equation whose unknown is a function and which must hold for all values of some variables. Both sides are built by composing the unknown function with known operations. Classic examples are Cauchy's functional equation $f(x + y) = f(x) + f(y)$ and the rule $\Gamma(x + 1) = x\,\Gamma(x)$ of the gamma function. By convention, equations with derivatives or integrals of the unknown are called differential or integral equations instead. Functional equations characterize many classical functions, such as the exponential, the logarithm and the power functions, by a few simple rules.