606
B Review of Traditional Calculus
properties that are shared by all real numbers, and they satisfy the exponent law
L m L n = L m+n . The commutation of D and J serves to illustrate these properties:
we have
DJ f (x) ≡
d
dx
x+h
x
f (t) dt = f (x + h) − f (x) =
x+h
x
df (t)
dt
dt = J Df (c) ,
from which we also may note that the operators J , D, and are interrelated through
J D = DJ = .
The proper inverse operator, L −1 , of an operator L is related to L by L −1 L =
LL −1 = 1. Some caution, however, does need to be applied to the inverses of , J ,
and D, as they annihilate, respectively, functions of period h, derivatives of functions
of period h, and constants. As we shall not normally be concerned here with
functions of period h, the only inverse operator with which we shall need to exercise
caution is D −1 . In this case, the cautionary note leads to the well-known ‘constant of
integration’ associated with antiderivatives, and discussed in every first year calculus
course. Stated otherwise, D −1 corresponds to the formation of an indefinite integral
(or antiderivative) and, while the derivative of the antiderivative gives precisely the
original function, the antiderivative of the derivative of that function introduces an
arbitrary additive constant, called the ‘constant of integration’.
In order to avoid this problem, we may note that D −1 f (x) is uniquely
determined, as the difference operator annihilates the arbitrary constant. From
the expression D −1 f (x) = F (x) + C, we see that the application of gives
D
−1 f (x) = F (x + h) − F (x) = J f (x) ,
which means that an equivalent operator equation is D −1 = J . We can also see
that the shifting operator, E, the difference operator D, and the integral operator, J ,
are related via DJ = J D = E − 1.
B.1.4 Taylor and Maclaurin Series
We shall have frequent need to make use of these concepts. However, we shall not
need to go beyond their development in a first year calculus course. You will recall
that the fundamental expressions for Taylor and Maclaurin series expansion are
f (x) = f (a) + f
(a)(x − a) +
1
2!
f
(a)(x − a)
2
+ · · ·
(B.1.17)
for the Taylor series expansion of a function f (x) about the point x = a, and
f (x) = f (0) + f
(0)x +
1
2!
f
(0)x
2
+ · · ·
(B.1.18)
B Review of Traditional Calculus
properties that are shared by all real numbers, and they satisfy the exponent law
L m L n = L m+n . The commutation of D and J serves to illustrate these properties:
we have
DJ f (x) ≡
d
dx
x+h
x
f (t) dt = f (x + h) − f (x) =
x+h
x
df (t)
dt
dt = J Df (c) ,
from which we also may note that the operators J , D, and are interrelated through
J D = DJ = .
The proper inverse operator, L −1 , of an operator L is related to L by L −1 L =
LL −1 = 1. Some caution, however, does need to be applied to the inverses of , J ,
and D, as they annihilate, respectively, functions of period h, derivatives of functions
of period h, and constants. As we shall not normally be concerned here with
functions of period h, the only inverse operator with which we shall need to exercise
caution is D −1 . In this case, the cautionary note leads to the well-known ‘constant of
integration’ associated with antiderivatives, and discussed in every first year calculus
course. Stated otherwise, D −1 corresponds to the formation of an indefinite integral
(or antiderivative) and, while the derivative of the antiderivative gives precisely the
original function, the antiderivative of the derivative of that function introduces an
arbitrary additive constant, called the ‘constant of integration’.
In order to avoid this problem, we may note that D −1 f (x) is uniquely
determined, as the difference operator annihilates the arbitrary constant. From
the expression D −1 f (x) = F (x) + C, we see that the application of gives
D
−1 f (x) = F (x + h) − F (x) = J f (x) ,
which means that an equivalent operator equation is D −1 = J . We can also see
that the shifting operator, E, the difference operator D, and the integral operator, J ,
are related via DJ = J D = E − 1.
B.1.4 Taylor and Maclaurin Series
We shall have frequent need to make use of these concepts. However, we shall not
need to go beyond their development in a first year calculus course. You will recall
that the fundamental expressions for Taylor and Maclaurin series expansion are
f (x) = f (a) + f
(a)(x − a) +
1
2!
f
(a)(x − a)
2
+ · · ·
(B.1.17)
for the Taylor series expansion of a function f (x) about the point x = a, and
f (x) = f (0) + f
(0)x +
1
2!
f
(0)x
2
+ · · ·
(B.1.18)