Definition of .
which makes sense whenever is differentiable (at least at ). However, the expression
makes sense even without being continuous, as long as . The expression is called a finite difference. The simplest case when , written
is called the forward difference of . For other non-zero , we write
When , it is called a backward difference of , sometimes written . Given a function and a real number , if we define and , then we have
Conversely, given and , we can find such that .
Some Properties of .
It is easy to see that the forward difference operator is linear:
, where is a constant.
also has the properties
The behavior of in this respect is similar to that of the derivative operator. However, because the continuity of is not assumed, does not imply that is a constant. is merely a periodic function . Other interesting properties include
for any real number
, where is the Bernoulli polynomial of order .
From , we can also form other operators. For example, we can iteratively define
Of course, all of the above can be readily generalized to . It is possible to show that can be written as a linear combination of
where is a one-dimensional real-valued function of . When 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.
|Date of creation||2013-03-22 15:35:00|
|Last modified on||2013-03-22 15:35:00|
|Last modified by||CWoo (3771)|