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µ µ
If f(x) ∈ AC, then the derivatives D a+ f(x) and D f(x), 0 < µ < 1, exist almost everywhere,
b–
µ µ
and we have D a+ f(x) ∈ L r (a, b) and D f(x) ∈ L r (a, b), 1 ≤ r <1/µ. These derivatives have the
b–
representations
x

1 f(a) f (t)
µ t
D f(x)= + dt , (7)
a+
Γ(1 – µ) (x – a) µ (x – t) µ
a
1 f(b) b f (t)

µ t
D f(x)= – dt . (8)
b– µ µ
Γ(1 – µ) (b – x) x (t – x)
Finally, let us pass to the fractional derivatives of order µ ≥ 1. We shall use the following
notation: [µ] stands for the integral part of a real number µ and {µ} is the fractional part of µ,
0 ≤{µ} < 1, so that
µ =[µ]+ {µ}. (9)
If µ is an integer, then by the fractional derivative of order µ we mean the ordinary derivative
µ µ

µ d µ d
D a+ = , D b– = – , µ =1, 2, ... (10)
dx dx
µ µ
However, if µ is not integral, then D a+ f and D f are introduced by the formulas
b–
[µ] [µ]+1
d d
D f(x) ≡ D {µ} f(x)= I 1–{µ} f(x), (11)
µ
a+
a+
a+
dx dx
[µ] [µ]+1
d d
D f(x) ≡ – D {µ} f(x)= – I 1–{µ} f(x). (12)
µ
b– b– b–
dx dx
Thus,
n
x
1 d f(t)
µ
D f(x)= dt, n =[µ] + 1, (13)
a+
Γ(n – µ) dx (x – t) µ–n+1
a
n
(–1) n d b f(t)
µ
D f(x)= dt, n =[µ] + 1. (14)
b– µ–n+1
Γ(n – µ) dx x (t – x)
A sufficient condition for the existence of the derivatives (13) and (14) is as follows:
x
f(t) dt [µ]

(x – t) {µ} ∈ AC .
a
This sufficient condition holds whenever f(x) ∈ AC [µ] .
Remark. The definitions of the fractional integrals and fractional derivatives can be extended to
the case of complex µ (e.g., see S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993)).
8.5-3. Main Properties
µ
Let I a+ (L 1 ), µ > 0, be the class of functions f(x) that can be represented by the left fractional integral
µ
of order µ of an integrable function: f(x)= I a+ ϕ(x), ϕ(x) ∈ L 1 (a, b), 1 ≤ p < ∞.
µ
For the relation f(x) ∈ I a+ (L 1 ), µ > 0, to hold, it is necessary and sufficient that
n
f n–µ (x) ≡ I n–µ f ∈ AC , (15)
a+
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 450