Fractional Calculus A Commutative Method on Real Analytic Functions

Fractional Calculus: A Commutative Method on Real Analytic Functions
Matthew Parker
Abstract
The traditional ﬁrst approach to fractional calculus is via the Riemann-
Liouville diﬀerintegral aDk
x [1]. The intent of this paper will be to create a
space K, pair of maps g : Cω
(R) → K and g : K → Cω
(R), and operator
Dk
: K → K such that the operator Dk
commutes with itself, the map g
embeds Cω
(R) isomorphically into K, and the following diagram commutes;
Cω
(R)
aDk
x

g
// K
Dk

Cω
(R) K
g
oo
This implies the following diagram commutes, for analytic f such that
aDj
xf = 0 (i.e, if f = i∈I bi(x-a)i
, where {bi} ⊂ R, and I ⊆ {j − 1, ..., j −
j });
f
aDj+k
x
##
aDj
x

g
// g(f)
Dj

0 Dj
g(f)
g
oo
Dk

aDj+k
x f Dk
Dj
g(f)
g
oo
Convention
Henceforth, unless otherwise noted we assume all functions are real-
analytic, thus equal to their Taylor series on some interval of R. When a
base point for a Taylor series is not given, we assume it converges on R or
the function has been analytically continued. We let Cω
(R) denote the space
of real analytic functions.
1
arXiv:1207.6610v1[math.CA]27Jul2012

The Space Zω(a)
From basic real analysis, for any f ∈ Cω
(R), the Taylor series of f
(henceforth denoted T(f)), equal to f on some open interval in R, is de-
fined by T(f) = ∞
i=0
f(i)(a)
i!
(x − a)i
for some a ∈ R. Note the collection
{f(i)
(a) : i ∈ Z≥0} together with the point a uniquely define f within
Cω
(R). Then, for fixed a ∈ R, there is a natural bijection between func-
tions σ : Z≥0 → R such that ∞
i=0
σ(i)
i!
(x − a)i
converges on some interval
about a, and functions f ∈ Cω
(R) equal to their Taylor series on some inter-
val about a.
Define Zω(a) to be the set of all functions σ : Z → R such that
∞
i=0
σ(i)
i!
(x − a)i
converges on some interval about a. When the point a
is understood, or not central to the argument but assumed to be fixed, we
may omit it and just write Zω. Note Zω is non-empty, since the function
f : i → 0 (all i ∈ Z) is an element of Zω. Moreover, Zω is a vector space
with identity 1ω : i → 0; if σ, ρ ∈ Zω and k ∈ R, ∞
i=0
(σ+ρ)(i)
i!
(x − a)i
=
∞
i=0
σ(i)
i!
(x − a)i
+ ∞
i=0
ρ(i)
i!
(x − a)i
∈ Cω
(R), and ∞
i=0
kσ(i)
i!
(x − a)i
=
k ∞
i=0
σ(i)
i!
(x − a)i
∈ Cω
(R)
The R Operator
For σ ∈ Zω(a), define the operator R : Zω(a) → Cω
(R) by Rσ =
∞
i=0
σ(i)
i!
(x − a)i
. Clearly R is surjective, with kernel {σ ∈ Zω : σ(i) =
0, i ≥ 0}. For some pair (f, a) ∈ Cω
(R) × R, we also define the operator
R−1
: Cω
(R) × R → Zω(a) by
R−1
(f, a)(i) =
0 : i 0
f(i)
(a) : i ≥ 0
This allows the following identities;
(R1) RR−1
(f, a) = (f, a)
(R1’) RR−1
f = f
(R2) R−1
Rσ(i) =
σ(i) i ≥ 0
0 i 0
By definition of the maps R and
R−1
, it follows they are both homomorphisms, where R−1
is injective with
image Zω/ker(R), and R is surjective.

The D-Operator and Γ Function
Given σ ∈ Zω(a), we define Dk
σ(i) = σ(i + k) for all k ∈ Z. From the
definition of the operator D, we immediately have the identities
(D1) Da
Db
= Db
Da
(D2) Da
Db
= Da+b
(D3) Da
D−a
= D−a
Da
= D0
(D4) Da
(σ + ρ) = Da
σ + Da
ρ
(D5) Da
(kσ) = kDa
σ for all k ∈ R
Relating the D Operator to Differentiation
Induction on the power rule provides the identity da
dxa xk
= k!
(k−a)!
xk−a
,
and the relation n! = Γ(n+1) provides the identity da
dxa xk
= Γ(k+1)
Γ(k+1−a)
xk−a
.
Applying these to Taylor series, we obtain the identity
T(f) =
∞
i=0
f(i)
(a)
i!
(x − a)i
=
∞
i=0
fi
(a)
Γ(i + 1)
(x − a)i
while the power rule allows for the identities
dj
dxj
T(f) = T(
dj
dxj
f) =
∞
i=0
i!
(i − j)!
f(i)
(a)(x−a)i−j
=
∞
i=0
Γ(i + 1)
Γ(i + 1 − j)
f(i)
(a)(x−a)i
For f ∈ Cω
(R) and σ ∈ Zω such that Rσ = f, straightforward calcula-
tion yields the following identities for all a ∈ Z;
(D6) RDa
R−1
f = da
dxa f = f(a)
(D7) R−1 da
dxa Rσ(i) =
σ(i + a) : i ≥ −a
0 : i − a
(D8) da
dxa RD−a
σ = f
Together, (D1) - (D8), along with (R1’) and (R2) will form the core of our
arguments for the rest of the paper.
Finally, we slightly redefine the operator R based on properties of the Γ
function. By definition, Rσ = ∞
i=0
σ(i)
Γ(i+1)
(x − a)i
. However, for i ≤ 0, σ(i)
Γ(i+1)
= 0 so σ(i)
Γ(i+1)
(x−a)i
= 0 and ∞
i=−∞
σ(i)
Γ(i+1)
(x−a)i
= ∞
i=0
σ(i)
Γ(i+1)
(x−a)i
= Rσ,
so from this point on we will define

Rσ =
∞
i=−∞
σ(i)
Γ(i + 1)
(x − a)i
Clearly, properties (R1), (R1’), (R2) and (D1) - (D8) still hold.
Mapping Cω
(R) To Rω And Back
For a power function b(x − s)α
, the Riemann-Liouville derivative aDk
x is
given by [1]
sDk
xb(x−s)α
=
b
Γ(−k)
x
s
(t−s)α
(x−t)−k−1
dt =
Γ(α + 1)
Γ(α + 1 − k)
b(x−s)α−k
If, and only if, k /∈ R and α + 1 − k ∈ Z≤0, then the numerator of
the fraction Γ(α+1)
Γ(α+1−k)
is finite while the denominator goes to ±∞, and in
the limit we see sDk
xf = 0. This shows that, when restricted to Cω
(R),
ker(sDk
x) = {b(x − s)α
: b, α ∈ R, α + 1 − k ∈ Z≤0}. If we wish to preserve
the identities (D1) - (D8), (R1’) and (R2’) when generalizing Dk
to all real
k, we must define a new operator with a significantly smaller kernel. Note
that (D1) - (D8) and (R1’), (R2) are only consistent if ker(Dk
) = {0}, the
zero function in Zω.
To summarize the situation, then, on the one hand we have the (com-
mutative) D operator on elements of Zω which, when coupled with the R op-
erator, allows identities (D1) - (D8), and on the other we have the Riemann-
Liouville derivative, which is commutative for analytic functions when the
degrees of differentiation under consideration never sum to a nonpositive
integer [2].
We now create maps f, f’ and a generalization of the D operator to a
space Rω (to be defined) such that the following diagram commutes;
Cω
(R)
f
//
aDk
x

Rω
Dk

Cω
(R) Rω
f
oo
It will, however, be more convenient to express the map f as a com-
position of maps Cω
(R)
R−1
−→ Zω
ι
−→ Rω and extending the domain of the
operator R to Rω so f = ι ◦ R−1
and f = R. Our goal, then, will be to de-
fine the maps and spaces which make the following diagram commute, while
maintaining analogs of (R1’) and (R2);

Cω
(R)
R−1
//
aDk
x

Zω
ι // Rω
Dk

Cω
Rω
Roo
Define Rω(a) = {ρ : R → R : ∞
i=−∞
Γ(i+1−k)
Γ(i+1)
ρ(i)(x−a)i−k
∈ Cω
(R)∀k ∈
R}. By definition, for any ρ ∈ Rω, ρ|Z
∈ Zω. In fact, for any ρ = ρ(x) ∈
Rω, ρ(x − k)|Z
∈ Zω. Observing D is merely a shift operator on Zω(R), we
naturally extend D to Rω by setting Dk
ρ(i) = ρ(i-k) for all k ∈ R, ρ ∈ Rω.
By definition of Rω, (Dk
ρ)|Z
∈ Zω for all k. This leads to the natural exten-
sion of R to Rω by Rρ = R(ρ|Z
) = ∞
i=−∞
ρ(i)
Γ(i+1)
(x − a)i
.
Properties (D1) - (D5) still hold for D on Rω, since elements of Rω, like
those of Zω, are functions. Thus, we are only left to define the map ι and
verify its properties.
Let σ ∈ Zω, then define ι(σ)(z) = limk→∞(aDz+k
x RD−k
σ)(a) whenever
the limit exists. We then have the following (equivalent) identities;
(I1) ι(σ)|Z
= σ
(I2) (Dk
ι(σ))|Z
= Dk
σ when k ∈ Z
(I3) ι(R−1
f)|Z
= R−1
f
(I4) R(ι(R−1
f)|Z
) = f
Identity (I4) is our analog of (R2), and (R1’) follows from properties
of R and (I1). Finally, we will show Diagram 2 commutes; that is, aDk
xf =
RDk
ι(R−1
f) for all k ∈ R. Let f ∈ Cω
(R), and fk : R → R be defined by
fk(z) = (aDz+k
x (RR−1
f))(a), then
RDk
ι(R−1
f) = Rι(R−1
f − k)
= R(fk)
=
∞
i=0
1
Γ(i + 1)
(aD−i+k
x f)(a)(x − a)i
= T(aDk
xf)
=a Dk
xf

and the diagram commutes. Then (D7) and the following analogs of (D6)
and (D8) hold;
(D6’) RDk
ι(R−1
f) =a Dk
xf
(D8’) aDk
xRD−k
ι(R−1
f) = f.
Conclusion
In conclusion, we have created a space Rω, and a collection of maps
and operators R, R−1
, Dk
, and ι such that the operator Dk
acts exactly the
same as the Riemann-Liouville operator as sDk
x when applied to an element
of Cω
(R) mapped through Rω, and the operator Dk
commutes with itself.
That is, if f ∈ Cω
(R) is such that sDj
xf = 0 for some j ∈ R, then for
σ = R−1
f, ρ = ι(σ), and all k ∈ R, we have the following commutative
diagram
f
aDj+k
x
##
aDj
x

R−1
// σ ι // ρ
Dj

0 Dk
ρR
oo
Dk

aDj+k
x f Dj
Dk
ρR
oo
which is equivalent to the second diagram in the abstract. This, together
with the second diagram in this section - which is equivalent to the first
diagram in the abstract - completes the paper.
References
[1] Keith B. Oldham and Jerome Spanier, The Fractional Calculus : Theory
and Applications of Differentiation and Integration to Arbitrary Order,
Dover, Mineola, New York, 2006.
[2] Kenneth S. Miller, An Introduction to the Fractional Calculus and Frac-
tional Differential Equations, Wiley-Interscience, 1993.

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