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39: Finite differences and functional equations


Functional equations are those in which a function is sought which is to satisfy certain relations among its values at all points. For example, we may look for functions satisfying f(x*y)=f(x)+f(y) and enquire whether the logarithm function f(x)=log(x) is the only solution. (It's not.) In some cases the nature of the answer is different when we insist that the functional equation hold for all real x, or all complex x, or only those in certain domains, for example.

A special case involves difference equations, that is, equations comparing f(x) - f(x-1), for example, with some expression involving x and f(x). In some ways these are discrete analogues of differential equations; thus we face similar questions of existence and uniqueness of solutions, global behaviour, and computational stability.


Applications and related fields

When the focus of a functional equation is on continuity of functions and a domain is specified, this becomes a question of topology (in particular this sometimes becomes questions about the group of homeomorphism or diffeomorphisms of a set. Thus see the manifolds page, for example.)

Functions whose domains are integers are sequences, of course; thus a functional equation with this domain is essentially a recursion problem. These are frequently seen among sequences of integers.

Functional equations are often studied by considering the orbits of points in the domain under iterates of some function. This then becomes the purview of dynamical systems (58FXX) .

Functions of one variable which satisfy a difference equation will tend to follow patterns set by ordinary differential equations; naturally functions of two or more variables behave more like solutions of partial differential equations. [Schematic of subareas and related areas]


There are only two subfields, which are then further subdivided:

This is among the smaller areas in the Math Reviews database.

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Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org