Generalization in fractional calculus
In mathematics , the Caputo fractional derivative , also called Caputo-type fractional derivative, is a generalization of derivatives for non-integer orders named after Michele Caputo. Caputo first defined this form of fractional derivative in 1967.[ 1]
The Caputo fractional derivative is motivated from the Riemann–Liouville fractional integral . Let
f
{\textstyle f}
(
0
,
∞
)
{\displaystyle \left(0,\,\infty \right)}
RL
I
{\textstyle {^{\text{RL}}\operatorname {I} }}
0
RL
I
x
α
[
f
(
x
)
]
=
1
Γ
(
−
α
)
⋅
∫
0
x
f
(
t
)
(
x
−
t
)
1
−
α
d
t
{\displaystyle {_{0}^{\text{RL}}\operatorname {I} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f\left(t\right)}{\left(x-t\right)^{1-\alpha }}}\,\operatorname {d} t}
where
Γ
(
⋅
)
{\textstyle \Gamma \left(\cdot \right)}
Gamma function .
Let's define
D
x
α
:=
d
α
d
x
α
{\textstyle \operatorname {D} _{x}^{\alpha }:={\frac {\operatorname {d} ^{\alpha }}{\operatorname {d} x^{\alpha }}}}
D
x
α
D
x
β
=
D
x
α
+
β
{\textstyle \operatorname {D} _{x}^{\alpha }\operatorname {D} _{x}^{\beta }=\operatorname {D} _{x}^{\alpha +\beta }}
D
x
α
=
RL
I
x
−
α
{\textstyle \operatorname {D} _{x}^{\alpha }={^{\text{RL}}\operatorname {I} _{x}^{-\alpha }}}
α
=
m
+
z
∈
R
∧
m
∈
N
0
∧
0
<
z
<
1
{\textstyle \alpha =m+z\in \mathbb {R} \wedge m\in \mathbb {N} _{0}\wedge 0<z<1}
D
x
α
=
D
x
m
+
z
=
D
x
z
+
m
=
D
x
z
−
1
+
1
+
m
=
D
x
z
−
1
D
x
1
+
m
=
RL
I
x
1
−
z
D
x
1
+
m
{\textstyle \operatorname {D} _{x}^{\alpha }=\operatorname {D} _{x}^{m+z}=\operatorname {D} _{x}^{z+m}=\operatorname {D} _{x}^{z-1+1+m}=\operatorname {D} _{x}^{z-1}\operatorname {D} _{x}^{1+m}={^{\text{RL}}\operatorname {I} }_{x}^{1-z}\operatorname {D} _{x}^{1+m}}
f
{\displaystyle f}
C
m
(
0
,
∞
)
{\displaystyle C^{m}\left(0,\,\infty \right)}
D
x
m
+
z
[
f
(
x
)
]
=
1
Γ
(
1
−
z
)
⋅
∫
0
x
f
(
1
+
m
)
(
t
)
(
x
−
t
)
z
d
t
.
{\displaystyle {\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(1+m\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t.}
This is known as the Caputo-type fractional derivative, often written as
C
D
x
α
{\textstyle {^{\text{C}}\operatorname {D} }_{x}^{\alpha }}
The first definition of the Caputo-type fractional derivative was given by Caputo as:
C
D
x
m
+
z
[
f
(
x
)
]
=
1
Γ
(
1
−
z
)
⋅
∫
0
x
f
(
m
+
1
)
(
t
)
(
x
−
t
)
z
d
t
{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(m+1\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t}
where
C
m
(
0
,
∞
)
{\displaystyle C^{m}\left(0,\,\infty \right)}
m
∈
N
0
∧
0
<
z
<
1
{\textstyle m\in \mathbb {N} _{0}\wedge 0<z<1}
[ 2]
A popular equivalent definition is:
C
D
x
α
[
f
(
x
)
]
=
1
Γ
(
⌈
α
⌉
−
α
)
⋅
∫
0
x
f
(
⌈
α
⌉
)
(
t
)
(
x
−
t
)
α
+
1
−
⌈
α
⌉
d
t
{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}
where
α
∈
R
>
0
∖
N
{\textstyle \alpha \in \mathbb {R} _{>0}\setminus \mathbb {N} }
⌈
⋅
⌉
{\textstyle \left\lceil \cdot \right\rceil }
ceiling function . This can be derived by substituting
α
=
m
+
z
{\textstyle \alpha =m+z}
⌈
α
⌉
=
m
+
1
{\textstyle \left\lceil \alpha \right\rceil =m+1}
⌈
α
⌉
+
z
=
α
+
1
{\textstyle \left\lceil \alpha \right\rceil +z=\alpha +1}
[ 3]
Another popular equivalent definition is given by:
C
D
x
α
[
f
(
x
)
]
=
1
Γ
(
n
−
α
)
⋅
∫
0
x
f
(
n
)
(
t
)
(
x
−
t
)
α
+
1
−
n
d
t
{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(n-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(n\right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-n}}}\,\operatorname {d} t}
where
n
−
1
<
α
<
n
∈
N
.
{\textstyle n-1<\alpha <n\in \mathbb {N} .}
The problem with these definitions is that they only allow arguments in
(
0
,
∞
)
{\textstyle \left(0,\,\infty \right)}
a
{\textstyle a}
a
C
D
x
α
[
f
(
x
)
]
=
1
Γ
(
⌈
α
⌉
−
α
)
⋅
∫
a
x
f
(
⌈
α
⌉
)
(
t
)
(
x
−
t
)
α
+
1
−
⌈
α
⌉
d
t
{\textstyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}
(
a
,
∞
)
{\textstyle \left(a,\,\infty \right)}
[ 4]
Properties and theorems [ edit ] Basic properties and theorems [ edit ] A few basic properties are:[ 5]
A table of basic properties and theorems
Properties
f
(
x
)
{\displaystyle f\left(x\right)}
a
C
D
x
α
[
f
(
x
)
]
{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]}
Condition
Definition
f
(
x
)
{\displaystyle f\left(x\right)}
f
(
α
)
(
x
)
−
f
(
α
)
(
a
)
{\displaystyle f^{\left(\alpha \right)}\left(x\right)-f^{\left(\alpha \right)}\left(a\right)}
Linearity
b
⋅
g
(
x
)
+
c
⋅
h
(
x
)
{\displaystyle b\cdot g\left(x\right)+c\cdot h\left(x\right)}
b
⋅
a
C
D
x
α
[
g
(
x
)
]
+
c
⋅
a
C
D
x
α
[
h
(
x
)
]
{\displaystyle b\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[g\left(x\right)\right]+c\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[h\left(x\right)\right]}
Index law
D
x
β
{\displaystyle \operatorname {D} _{x}^{\beta }}
a
C
D
x
α
+
β
{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha +\beta }}}
β
∈
Z
{\displaystyle \beta \in \mathbb {Z} }
Semigroup property
a
C
D
x
β
{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\beta }}}
a
C
D
x
α
+
β
{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha +\beta }}}
⌈
α
⌉
=
⌈
β
⌉
{\displaystyle \left\lceil \alpha \right\rceil =\left\lceil \beta \right\rceil }
The index law does not always fulfill the property of commutation:
a
C
D
x
α
a
C
D
x
β
=
a
C
D
x
α
+
β
≠
a
C
D
x
β
a
C
D
x
α
{\displaystyle \operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }\operatorname {_{a}^{\text{C}}D} _{x}^{\beta }=\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha +\beta }\neq \operatorname {_{a}^{\text{C}}D} _{x}^{\beta }\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }}
where
α
∈
R
>
0
∖
N
∧
β
∈
N
{\displaystyle \alpha \in \mathbb {R} _{>0}\setminus \mathbb {N} \wedge \beta \in \mathbb {N} }
Fractional Leibniz rule [ edit ] The Leibniz rule for the Caputo fractional derivative is given by:
a
C
D
x
α
[
g
(
x
)
⋅
h
(
x
)
]
=
∑
k
=
0
∞
[
(
a
k
)
⋅
g
(
k
)
(
x
)
⋅
a
RL
D
x
α
−
k
[
h
(
x
)
]
]
−
(
x
−
a
)
−
α
Γ
(
1
−
α
)
⋅
g
(
a
)
⋅
h
(
a
)
{\displaystyle \operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }\left[g\left(x\right)\cdot h\left(x\right)\right]=\sum \limits _{k=0}^{\infty }\left[{\binom {a}{k}}\cdot g^{\left(k\right)}\left(x\right)\cdot \operatorname {_{a}^{\text{RL}}D} _{x}^{\alpha -k}\left[h\left(x\right)\right]\right]-{\frac {\left(x-a\right)^{-\alpha }}{\Gamma \left(1-\alpha \right)}}\cdot g\left(a\right)\cdot h\left(a\right)}
where
(
a
b
)
=
Γ
(
a
+
1
)
Γ
(
b
+
1
)
⋅
Γ
(
a
−
b
+
1
)
{\textstyle {\binom {a}{b}}={\frac {\Gamma \left(a+1\right)}{\Gamma \left(b+1\right)\cdot \Gamma \left(a-b+1\right)}}}
[ 6] [ 7]
Relation to other fractional differential operators [ edit ] Caputo-type fractional derivative is closely related to the Riemann–Liouville fractional integral via its definition:
a
C
D
x
α
[
f
(
x
)
]
=
a
RL
I
x
⌈
α
⌉
−
α
[
D
x
⌈
α
⌉
[
f
(
x
)
]
]
{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[f\left(x\right)\right]\right]}
Furthermore, the following relation applies:
a
C
D
x
α
[
f
(
x
)
]
=
a
RL
D
x
α
[
f
(
x
)
]
−
∑
k
=
0
⌈
α
⌉
[
x
k
−
α
Γ
(
k
−
α
+
1
)
⋅
f
(
k
)
(
0
)
]
{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[{\frac {x^{k-\alpha }}{\Gamma \left(k-\alpha +1\right)}}\cdot f^{\left(k\right)}\left(0\right)\right]}
where
a
RL
D
x
α
{\displaystyle {_{a}^{\text{RL}}\operatorname {D} _{x}^{\alpha }}}
The Laplace transform of the Caputo-type fractional derivative is given by:
L
x
{
a
C
D
x
α
[
f
(
x
)
]
}
(
s
)
=
s
α
⋅
F
(
s
)
−
∑
k
=
0
⌈
α
⌉
[
s
α
−
k
−
1
⋅
f
(
k
)
(
0
)
]
{\displaystyle {\mathcal {L}}_{x}\left\{{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]\right\}\left(s\right)=s^{\alpha }\cdot F\left(s\right)-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[s^{\alpha -k-1}\cdot f^{\left(k\right)}\left(0\right)\right]}
where
L
x
{
f
(
x
)
}
(
s
)
=
F
(
s
)
{\textstyle {\mathcal {L}}_{x}\left\{f\left(x\right)\right\}\left(s\right)=F\left(s\right)}
[ 8]
Caputo fractional derivative of some functions [ edit ] The Caputo fractional derivative of a constant
c
{\displaystyle c}
a
C
D
x
α
[
c
]
=
1
Γ
(
⌈
α
⌉
−
α
)
⋅
∫
a
x
D
t
⌈
α
⌉
[
c
]
(
x
−
t
)
α
+
1
−
⌈
α
⌉
d
t
=
1
Γ
(
⌈
α
⌉
−
α
)
⋅
∫
a
x
0
(
x
−
t
)
α
+
1
−
⌈
α
⌉
d
t
a
C
D
x
α
[
c
]
=
0
{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {\operatorname {D} _{t}^{\left\lceil \alpha \right\rceil }\left[c\right]}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {0}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&=0\end{aligned}}}
The Caputo fractional derivative of a power function
x
b
{\displaystyle x^{b}}
[ 9]
a
C
D
x
α
[
x
b
]
=
a
RL
I
x
⌈
α
⌉
−
α
[
D
x
⌈
α
⌉
[
x
b
]
]
=
Γ
(
b
+
1
)
Γ
(
b
−
⌈
α
⌉
+
1
)
⋅
a
RL
I
x
⌈
α
⌉
−
α
[
x
b
−
⌈
α
⌉
]
a
C
D
x
α
[
x
b
]
=
{
Γ
(
b
+
1
)
Γ
(
b
−
α
+
1
)
(
x
b
−
α
−
a
b
−
α
)
,
for
⌈
α
⌉
−
1
<
b
∧
b
∈
R
0
,
for
⌈
α
⌉
−
1
≥
b
∧
b
∈
N
{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[x^{b}\right]\right]={\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\left\lceil \alpha \right\rceil +1\right)}}\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[x^{b-\left\lceil \alpha \right\rceil }\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={\begin{cases}{\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\alpha +1\right)}}\left(x^{b-\alpha }-a^{b-\alpha }\right),\,&{\text{for }}\left\lceil \alpha \right\rceil -1<b\wedge b\in \mathbb {R} \\0,\,&{\text{for }}\left\lceil \alpha \right\rceil -1\geq b\wedge b\in \mathbb {N} \\\end{cases}}\end{aligned}}}
The Caputo fractional derivative of a exponential function
e
a
⋅
x
{\displaystyle e^{a\cdot x}}
a
C
D
x
α
[
e
b
⋅
x
]
=
a
RL
I
x
⌈
α
⌉
−
α
[
D
x
⌈
α
⌉
[
e
b
⋅
x
]
]
=
b
⌈
α
⌉
⋅
a
RL
I
x
⌈
α
⌉
−
α
[
e
b
⋅
x
]
a
C
D
x
α
[
e
b
⋅
x
]
=
b
α
⋅
(
E
x
(
⌈
α
⌉
−
α
,
b
)
−
E
a
(
⌈
α
⌉
−
α
,
b
)
)
{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[e^{b\cdot x}\right]\right]=b^{\left\lceil \alpha \right\rceil }\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[e^{b\cdot x}\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&=b^{\alpha }\cdot \left(E_{x}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)-E_{a}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)\right)\\\end{aligned}}}
where
E
x
(
ν
,
a
)
=
a
−
ν
⋅
e
a
⋅
x
⋅
γ
(
ν
,
a
⋅
x
)
Γ
(
ν
)
{\textstyle E_{x}\left(\nu ,\,a\right)={\frac {a^{-\nu }\cdot e^{a\cdot x}\cdot \gamma \left(\nu ,\,a\cdot x\right)}{\Gamma \left(\nu \right)}}}
E
t
{\textstyle \operatorname {E} _{t}}
γ
(
a
,
b
)
{\textstyle \gamma \left(a,\,b\right)}
incomplete gamma function .[ 10]
^ Diethelm, Kai (2019). "General theory of Caputo-type fractional differential equations" . Fractional Differential Equations . pp. 1–20. doi :10.1515/9783110571660-001 . ISBN 978-3-11-057166-0 . Retrieved 2023-08-10 . ^ Caputo, Michele (1967). "Linear Models of Dissipation whose Q is almost Frequency Independent-II" . ResearchGate . 13 (5): 530. Bibcode :1967GeoJ...13..529C . doi :10.1111/j.1365-246X.1967.tb02303.x ^ Lazarević, Mihailo; Rapaić, Milan Rade; Šekara, Tomislav (2014). "Introduction to Fractional Calculus with Brief Historical Background" . ResearchGate : 8. ^ Dimitrov, Yuri; Georgiev, Slavi; Todorov, Venelin (2023). "Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations" . Fractal and Fractional . 7 (10): 750. doi :10.3390/fractalfract7100750 ^ Sikora, Beata (2023). "Remarks on the Caputo fractional derivative" (PDF) . Matematyka I Informatyka Na Uczelniach Technicznych (5): 78–79. ^ Huseynov, Ismail; Ahmadova, Arzu; Mahmudov, Nazim (2020). "Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications" . ResearchGate : 1. arXiv :2012.11360 ^ Weisstein, Eric W. (2024). "Binomial Coefficient" . mathworld.wolfram.com . Retrieved 2024-05-20 . ^ Sontakke, Bhausaheb Rajba; Shaikh, Amjad (2015). "Properties of Caputo Operator and Its Applications to Linear Fractional Differential Equations" (PDF) . Journal of Engineering Research and Applications . 5 (5): 23–24. ISSN 2248-9622 . ^ Weisstein, Eric W. "Fractional Derivative" . mathworld.wolfram.com . Retrieved 2024-05-20 . ^ Weisstein, Eric W. (2024). "E_t-Function" . mathworld.wolfram.com . Retrieved 2024-05-20 .
![{\displaystyle {_{0}^{\text{RL}}\operatorname {I} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f\left(t\right)}{\left(x-t\right)^{1-\alpha }}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd70814fda4cbee4288ca757850b88057e3dddc4)
![{\displaystyle {\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(1+m\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a081af9954ae43d6839b0cc4da29bb9c3d2a3f4f)
![{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(m+1\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/138f94697ffd5b346e2e43660224e77b9de1cae9)
![{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71777eade0c4722f4bacae90571d8da69bc512f7)
![{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(n-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(n\right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-n}}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ec7714bac06791e632ef971e979fc84c02bc2f)
![{\textstyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5535c35870b70feb48a7ac931d6c4e88b7be5c6)
![{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd8cacbd2e730ad68dbad7868e4f7e5a11f5d19c)
![{\displaystyle b\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[g\left(x\right)\right]+c\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[h\left(x\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c067167108b7a82c81284037102d8791b575b06f)
![{\displaystyle \operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }\left[g\left(x\right)\cdot h\left(x\right)\right]=\sum \limits _{k=0}^{\infty }\left[{\binom {a}{k}}\cdot g^{\left(k\right)}\left(x\right)\cdot \operatorname {_{a}^{\text{RL}}D} _{x}^{\alpha -k}\left[h\left(x\right)\right]\right]-{\frac {\left(x-a\right)^{-\alpha }}{\Gamma \left(1-\alpha \right)}}\cdot g\left(a\right)\cdot h\left(a\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12bf52380a489942d650c30cd48f0fa02cb0f60d)
![{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[f\left(x\right)\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31b0212b2f9973ffc903eb596d075cd1fdb2397c)
![{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[{\frac {x^{k-\alpha }}{\Gamma \left(k-\alpha +1\right)}}\cdot f^{\left(k\right)}\left(0\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac1591dd4516ee8f603666f9828d91158842ca4)
![{\displaystyle {\mathcal {L}}_{x}\left\{{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]\right\}\left(s\right)=s^{\alpha }\cdot F\left(s\right)-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[s^{\alpha -k-1}\cdot f^{\left(k\right)}\left(0\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12b929ca9b9401e6152a475f87fb04a30bfec5fb)
![{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {\operatorname {D} _{t}^{\left\lceil \alpha \right\rceil }\left[c\right]}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {0}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01dd0bb475bc01fa68ee24b5a24b635b2580b0a6)
![{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[x^{b}\right]\right]={\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\left\lceil \alpha \right\rceil +1\right)}}\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[x^{b-\left\lceil \alpha \right\rceil }\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={\begin{cases}{\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\alpha +1\right)}}\left(x^{b-\alpha }-a^{b-\alpha }\right),\,&{\text{for }}\left\lceil \alpha \right\rceil -1<b\wedge b\in \mathbb {R} \\0,\,&{\text{for }}\left\lceil \alpha \right\rceil -1\geq b\wedge b\in \mathbb {N} \\\end{cases}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f9f9d9bfc07939992110d7c4a38b85ff0f408ea)
![{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[e^{b\cdot x}\right]\right]=b^{\left\lceil \alpha \right\rceil }\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[e^{b\cdot x}\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&=b^{\alpha }\cdot \left(E_{x}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)-E_{a}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7d8409f8544e0b3f07907a4973db9ec168071e)
