Biomedical Engineering Reference
In-Depth Information
Applying this formula again, we can find the second derivative:
f
(x
f
(x)
+
h)
−
f
(x)
=
lim
h
h
→
0
f(x
+
h
1
+
h
2
)
−
f(x
+
h
1
)
h
2
f(x
+
h
2
)
−
f(x)
h
2
lim
h
2
→
0
−
lim
h
2
→
0
=
lim
h
1
→
(A.17)
h
1
0
By choosing the same value of
h
,i.e.
h
=
h
1
=
h
2
, the expression simplifies to
f(x
+
2
h)
−
2
f(x
+
h)
+
f(x)
f
(x)
=
lim
h
(A.18)
h
2
→
0
For the
n
th derivative, this procedure can be consolidated into a summation. We
introduce the operator
d
n
to represent the
n
-repetitions of the derivative:
1
)
m
n
m
f(x
−
mh)
n
1
h
n
d
n
f(x)
=
lim
h
→
(
−
(A.19)
0
m
=
0
R
pro-
vided that the binomial coefficient be understood as using the Gamma Function in
place of the standard factorial. Also, the upper limit of the summation (no longer
the integer,
n
) goes to infinity as
This expression can be generalized for non-integer values for
n
with
α
∈
t
−
1
h
(where
t
and a are the upper and lower lim-
its of differentiation, respectively). We are left with the generalized form of the
Grunwald-Letnikov fractional derivative.
t
−
a
h
1
h
n
(α
+
1
)
d
n
f(x)
1
)
m
=
lim
h
(
−
1
)
f(x
−
mh)
(A.20)
m
!
(α
−
m
+
→
0
m
=
0
It is obvious that, just as the Riemann-Liouville definition for the fractional in-
tegral could be used to define the fractional derivative, the above form of the GL
derivative could be altered for use in an alternate definition of the fractional integral.
The most natural alteration of this form is to consider the GL derivative for negative
α
. If we revert to the (
A.19
) form the most immediate problem is that (
−
m
)isnot
defined using factorials. Expanded mathematically, (
−
m
) is given by
−
n
m
=
−
n(
−
n
−
1
)(
−
n
−
2
)(
−
n
−
3
)
···
(
−
n
−
m
+
1
)
(A.21)
m
!
This form can be rewritten as
−
n
m
1
)
m
−
n(
−
n
−
1
)(
−
n
−
2
)(
−
n
−
3
)
···
(
−
n
−
m
+
1
)
=
(
−
m
!
+
−
!
1
)
m
(n
m
1
)
=
−
(
(A.22)
−
!
!
(n
1
)
m
