finite difference
Definition of $\mathrm{\Delta}$.
The derivative^{} of a function^{} $f:\mathbb{R}\to \mathbb{R}$ is defined to be the expression
$$\frac{df}{dx}:=\underset{h\to 0}{lim}\frac{f(x+h)f(x)}{h},$$ 
which makes sense whenever $f$ is differentiable^{} (at least at $x$). However, the expression
$$\frac{f(x+h)f(x)}{h}$$ 
makes sense even without $f$ being continuous^{}, as long as $h\ne 0$. The expression is called a finite difference. The simplest case when $h=1$, written
$$\mathrm{\Delta}f(x):=f(x+1)f(x),$$ 
is called the forward difference^{} of $f$. For other nonzero $h$, we write
$${\mathrm{\Delta}}_{h}f(x):=\frac{f(x+h)f(x)}{h}.$$ 
When $h=1$, it is called a backward difference of $f$, sometimes written $\nabla f(x):={\mathrm{\Delta}}_{1}f(x)$. Given a function $f(x)$ and a real number $h\ne 0$, if we define $y=\frac{x}{h}$ and $g(y)=\frac{f(hy)}{h}$, then we have
$$\mathrm{\Delta}g(y)={\mathrm{\Delta}}_{h}f(x).$$ 
Conversely, given $g(y)$ and $h\ne 0$, we can find $f(x)$ such that $\mathrm{\Delta}g(y)={\mathrm{\Delta}}_{h}f(x)$.
Some Properties of $\mathrm{\Delta}$.
It is easy to see that the forward difference operator $\mathrm{\Delta}$ is linear:

1.
$\mathrm{\Delta}(f+g)=\mathrm{\Delta}(f)+\mathrm{\Delta}(g)$

2.
$\mathrm{\Delta}(cf)=c\mathrm{\Delta}(f)$, where $c\in \mathbb{R}$ is a constant.
$\mathrm{\Delta}$ also has the properties

1.
$\mathrm{\Delta}(c)=0$ for any realvalued constant function^{} $c$, and

2.
$\mathrm{\Delta}(I)=1$ for the identity function^{} $I(x)=x$. constant.
The behavior of $\mathrm{\Delta}$ in this respect is similar^{} to that of the derivative operator. However, because the continuity of $f$ is not assumed, $\mathrm{\Delta}f=0$ does not imply that $f$ is a constant. $f$ is merely a periodic function $f(x+1)=f(x)$. Other interesting properties include

1.
$\mathrm{\Delta}{a}^{x}=(a1){a}^{x}$ for any real number $a$

2.
$\mathrm{\Delta}{x}^{(n)}=n{x}^{(n1)}$ where ${x}^{(n)}$ denotes the falling factorial^{} polynomial^{}

3.
$\mathrm{\Delta}{b}_{n}(x)=n{x}^{n1}$, where ${b}_{n}(x)$ is the Bernoulli polynomial^{} of order $n$.
From $\mathrm{\Delta}$, we can also form other operators. For example, we can iteratively define
${\mathrm{\Delta}}^{1}f:=\mathrm{\Delta}f$  (1)  
${\mathrm{\Delta}}^{k}f:=\mathrm{\Delta}({\mathrm{\Delta}}^{k1}f),\text{where}k1.$  (2) 
Of course, all of the above can be readily generalized to ${\mathrm{\Delta}}_{h}$. It is possible to show that ${\mathrm{\Delta}}_{h}f$ can be written as a linear combination^{} of
$$\mathrm{\Delta}f,{\mathrm{\Delta}}^{2}f,\mathrm{\dots},{\mathrm{\Delta}}^{h}f.$$ 
Suppose $F:{\mathbb{R}}^{n}\to \mathbb{R}$ is a realvalued function whose domain is the $n$dimensional Euclidean space. A difference equation (in one variable $x$) is the equation of the form
$$F(x,{\mathrm{\Delta}}_{{h}_{1}}^{{k}_{1}}f,{\mathrm{\Delta}}_{{h}_{2}}^{{k}_{2}}f,\mathrm{\dots},{\mathrm{\Delta}}_{{h}_{n}}^{{k}_{n}}f)=0,$$ 
where $f:=f(x)$ is a onedimensional realvalued function of $x$. When ${h}_{i}$ are all integers, the expression on the left hand side of the difference equation can be rewritten and simplified as
$$G(x,f,\mathrm{\Delta}f,{\mathrm{\Delta}}^{2}f,\mathrm{\dots},{\mathrm{\Delta}}^{m}f)=0.$$ 
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  20130322 15:35:00 
Last modified on  20130322 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 