Next: Differentiation Formulas Up:Main Previous: Difference Notation
For many purposes, it is convenient to think of the symbols
and
defined earlier, as operators,
which transform a given function
into related functions,
according to the laws:
In addition, we also define the following:
Averaging operator , denoted by and
defined as
Shift operator, denoted by , and defined as
and the differential operator, denoted by
and defined as
In all these operators except D, the spacing h is implied.
Positive integral powers of these operators are defined by
iteration. Also we define the zeroeth power of any operator as the
identity operator I, which leaves any function
unchanged. For the operator
, the power
is defined
for any
real so that
assuming the existence of
. In view of the above,
we also have
so that
Again
giving
Also
and thus
Moreover,
and thus
Hence the operators , , and , are
simply expressed in terms of . From the above relations, we may
deduce the relations
after which the formal symbolism of elementary algebra suggests
the forms
and
while the
first form requires no explanation, the form
can be interpreted at this stage only as
representing the inverse of operator
, that is, as an
alternative notation of the operator
such that
whereas the derivation of the third form shows that
is to represent an
operator such that its iterate is the
operator
,
Next:Differentiation Formulas
Up:Main Previous: Difference Notation