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# 39: Finite differences and functional equations

## Introduction

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.

## Subfields

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

• 39A: Difference equations, For dynamical systems, see 58FXX
• 39A05: General
• 39A10: Difference equations, See also 33Dxx
• 39A11: Stability of difference equations
• 39A12: Discrete version of topics in analysis
• 39A70: Difference operators, See also 47B39
• 39A99: None of the above but in this section
• 39B: Functional equations, see also 30D05
• 39B05: General
• 39B12: Iteration theory, iterative and composite equations, See also 26A18, 30D05, 58F08
• 39B22: Equations for real functions
• 39B32: Equations for complex functions, See also 30D05
• 39B42: Matrix and operator equations
• 39B52: Equations for functions with more general domains and/or ranges
• 39B62: Systems of functional equations
• 39B72: Inequalities involving unknown functions, See also 26A51, 26Dxx
• 39B99: None of the above but in this section

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

Browse all (old) classifications for this area at the AMS.

## Selected topics at this site

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