finite difference

Definition of Δ.

The derivativePlanetmathPlanetmath of a functionMathworldPlanetmath f: is defined to be the expression


which makes sense whenever f is differentiableMathworldPlanetmathPlanetmath (at least at x). However, the expression


makes sense even without f being continuousMathworldPlanetmathPlanetmath, as long as h0. The expression is called a finite difference. The simplest case when h=1, written


is called the forward differencePlanetmathPlanetmath of f. For other non-zero h, we write


When h=-1, it is called a backward difference of f, sometimes written f(x):=Δ-1f(x). Given a function f(x) and a real number h0, if we define y=xh and g(y)=f(hy)h, then we have


Conversely, given g(y) and h0, we can find f(x) such that Δg(y)=Δhf(x).

Some Properties of Δ.

It is easy to see that the forward difference operator Δ is linear:

  1. 1.


  2. 2.

    Δ(cf)=cΔ(f), where c is a constant.

Δ also has the properties

  1. 1.

    Δ(c)=0 for any real-valued constant functionMathworldPlanetmath c, and

  2. 2.

    Δ(I)=1 for the identity functionMathworldPlanetmath I(x)=x. constant.

The behavior of Δ in this respect is similarPlanetmathPlanetmath to that of the derivative operator. However, because the continuity of f is not assumed, Δf=0 does not imply that f is a constant. f is merely a periodic function f(x+1)=f(x). Other interesting properties include

  1. 1.

    Δax=(a-1)ax for any real number a

  2. 2.

    Δx(n)=nx(n-1) where x(n) denotes the falling factorialDlmfMathworld polynomialMathworldPlanetmathPlanetmathPlanetmath

  3. 3.

    Δbn(x)=nxn-1, where bn(x) is the Bernoulli polynomialDlmfDlmfPlanetmathPlanetmath of order n.

From Δ, we can also form other operators. For example, we can iteratively define

Δ1f:=Δf (1)
Δkf:=Δ(Δk-1f),where k>1. (2)

Of course, all of the above can be readily generalized to Δh. It is possible to show that Δhf can be written as a linear combinationMathworldPlanetmath of


Suppose F:n is a real-valued function whose domain is the n-dimensional Euclidean space. A difference equation (in one variable x) is the equation of the form


where f:=f(x) is a one-dimensional real-valued function of x. When hi are all integers, the expression on the left hand side of the difference equation can be re-written and simplified as


Difference equations are used in many problems in the real world, one example being in the study of traffic flow.

Title finite difference
Canonical name FiniteDifference
Date of creation 2013-03-22 15:35:00
Last modified on 2013-03-22 15:35:00
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 65Q05
Related topic Equation
Related topic RecurrenceRelation
Related topic IndefiniteSum
Related topic DifferentialPropositionalCalculus
Defines forward difference
Defines backward difference
Defines difference equation