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 all these operations except D, the spacing h is implied.
Positive integral power of these operators are defined by
iteration. Also we define the zeroth 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
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