functional equation

In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. Many properties of functions can be determined by studying the types of functional equations they satisfy. Usually the term functional equation is reserved for equations that are not in some simple sense reducible to algebraic equations, often because two or more known functions of the variables are substituted as arguments into an unknown function to be solved for.

Examples

The functional equation

\zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)

is satisfied by the Riemann zeta function ζ. The capital Γ denotes the gamma function.

The functional equation

x\Gamma(x)=\Gamma(x+1)

is satisfied by the gamma function.

The functional equation

f\left({az+b\over cz+d}\right) = (cz+d)^k f(z)

where a, b, c, d are integers satisfying ad − bc = 1, is involved in the definition of the concept of modular form.

Miscellaneous examples not necessarily involving "famous" functions:

f(x + y) = f(x)f(y), satisfied by all exponential functions

f(xy) = f(x) + f(y), satisfied by all logarithmic functions

f(x + y) = f(x) + f(y) (Cauchy equation)

F(az) = aF(z)(1 − F(z)) (Poincar equation)

G(x) = λ⁻¹ G(G(λz)) (chaos theory, scaling)

f((x + y)/2) = (f(x) + f(y))/2 (Jensen)

g(x + y) + g(x − y) = 2g(x)g(y) (d'Alembert)

f(h(x)) = cf(x) (Schrder)

f(h(x)) = f(x) + 1 (Abel).

A simple form of functional equation is a recurrence relation. This, formally speaking, involves an unknown function on integers, and also translation operators.

One such example of a recurrence relation is

a(n) = 3a(n-1) + 4a(n-2)

The commutative and associative laws are functional equations. When the associative law is expressed in its familiar form, one lets some symbol between two variables represent a binary operation, thus:

(a * b) * c = a * (b * c),

But if we write f(a, b) instead of a * b, then the associative law looks more like what one conventionally thinks of as a functional equation:

f(f(a, b), c) = f(a, f(b, c)).

One thing that all of the examples listed above share in common is that in each case two or more known functions (sometimes multiplication by a constant, sometimes addition of two variables, sometimes the identity function) are substituted into the unknown function to be solved for. When it comes to asking for all solutions, it may be the case that conditions from mathematical analysis should be applied; for example, in the case of the Cauchy equation mentioned above, the solutions that are continuous functions are the 'reasonable' ones, while other solutions that are not likely to have practical application can be constructed (by using a Hamel basis for the real numbers as vector space over the rational numbers). The Bohr-Mollerup theorem is another well-known example.

Functional Equation

Examples

See also