In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the logarithmic functional equation log ( x y ) = log ( x ) + log ( y ) {\displaystyle \log(xy)=\log(x)+\log(y)} . If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have highly irregular solutions. For example, the gamma function is a function that satisfies the functional equation f ( x + 1 ) = x f ( x ) {\displaystyle f(x+1)=xf(x)} and the initial value f ( 1 ) = 1. {\displaystyle f(1)=1.} There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for x real and positive (Bohr–Mollerup theorem).
Examples Recurrence relations can be seen as functional equations in functions over the integers or natural numbers, in which the differences between terms' indexes can be seen as an application of the shift operator. For example, the recurrence relation defining the Fibonacci numbers, F n = F n − 1 + F n − 2 {\displaystyle F_{n}=F_{n-1}+F_{n-2}} , where F 0 = 0 {\displaystyle F_{0}=0} and F 1 = 1 {\displaystyle F_{1}=1}
f ( x + P ) = f ( x ) {\displaystyle f(x+P)=f(x)} , which characterizes the periodic functions
f ( x ) = f ( − x ) {\displaystyle f(x)=f(-x)} , which characterizes the even functions, and likewise f ( x ) = − f ( − x ) {\displaystyle f(x)=-f(-x)} , which characterizes the odd functions
f ( f ( x ) ) = g ( x ) {\displaystyle f(f(x))=g(x)} , which characterizes the functional square roots of a function g {\displaystyle g}
f ( x + y ) = f ( x ) + f ( y ) {\displaystyle f(x+y)=f(x)+f(y)} (Cauchy's functional equation), satisfied by linear maps. The equation may, contingent on the axiom of choice, also have other pathological nonlinear solutions, whose existence can be proven with a Hamel basis for the real numbers
f ( x + y ) = f ( x ) f ( y ) , {\displaystyle f(x+y)=f(x)f(y),} satisfied by all exponential functions. Like Cauchy's additive functional equation, this too may have pathological, discontinuous solutions
f ( x y ) = f ( x ) + f ( y ) {\displaystyle f(xy)=f(x)+f(y)} , satisfied by all logarithmic functions and, over coprime integer arguments, additive functions
f ( x y ) = f ( x ) f ( y ) {\displaystyle f(xy)=f(x)f(y)} , satisfied by all power functions and, over coprime integer arguments, multiplicative functions
f ( x + y ) + f ( x − y ) = 2 [ f ( x ) + f ( y ) ] {\displaystyle f(x+y)+f(x-y)=2[f(x)+f(y)]} (quadratic equation or parallelogram law)
f ( ( x + y ) / 2 ) = ( f ( x ) + f ( y ) ) / 2 {\displaystyle f((x+y)/2)=(f(x)+f(y))/2} (Jensen's functional equation)
g ( x + y ) + g ( x − y ) = 2 [ g ( x ) g ( y ) ] {\displaystyle g(x+y)+g(x-y)=2[g(x)g(y)]} (d'Alembert's functional equation)
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