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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

Examples

• The functional equation

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

• The functional equation

xΓ(x) = Γ(x + 1)

is satisfied by the gamma function.

• The functional equation

where a, b, c, d are integers satisfying ad − bc = 1, defines f to be a modular form of order k.

• 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)) (Poincaré 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) (Schröder)

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. In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ... The Gamma function along an interval In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. ... The Gamma function along an interval In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ... The exponential function is one of the most important functions in mathematics. ... In mathematics, a logarithm of x with base b may be defined as the following: for the equation bn = x, the logarithm is a function which gives n. ... Recurrent redirects here; for the meaning of recurrent in contemporary hit radio, see Recurrent rotation. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In Euclidean geometry, a translation, or translation operator, is an affine transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...

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. Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematical analysis, the Bohr_Mollerup theorem, named after the Danish mathematicians Harald Bohr and Johannes Mollerup, who proved it, characterizes the gamma function, defined for x > 0 by as the only function f on the interval x > 0 that simultaneously has the three properties and and is a convex function. ...

External links

• Functional Equations: Exact Solutions (http://eqworld.ipmnet.ru/en/solutions/fe.htm) at EqWorld: The World of Mathematical Equations.
• Functional Equations: Index (http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-fe.htm) at EqWorld: The World of Mathematical Equations.

See also

• Functional equation (L-function)