Mathematical Physics Vol 1

Chapter A. Fractional Calculus: A Survey of Useful Formulas

460

R Liouville integrals and derivatives are Riemann-Liouville integrals and derivatives for the particular case a = − ∞ or b =+ ∞ ; that is to say, L I α ± f ( x ) def = RL I α ∓ ∞ ± f ( x ) and L D α ± f ( x ) def = RL D α ∓ ∞ ± f ( x ) . Sometimes LI α − is named the fractional integral of Weyl, and LD α − the Weyl transform, W α .

If D α is any fractional derivative, the Miller-Ross sequential derivative of order k α , k ∈ Z is givenby D α = D α , D k α = D α D ( k − 1 ) α . (A.37)

R

β f ( t ) is integrable, then:

R If f ( t ) has β : max { 0 , ⌊ α ⌋} continuous derivatives, and D

RL D α

α a ±

f ( x )= GL D

f ( x ) ,

(A.38)

a ±

a + " f ( u ) −

( u − a ) k #! ( x ) , ( b − u ) k #! ( x ) .

⌈ ℜ ( α ) ⌉− 1 ∑ k = 0 ⌈ ℜ ( α ) ⌉− 1 ∑ k = 0

f ( k ) ( a ) k ! f ( k ) ( a ) k !

C D α

RL D α

a + f ( x )=

(A.39)

b − "

C D α

f ( x )= RL D α

f ( u ) −

(A.40)

b −

Equations (A.39)-((A.40) are sometimes considered as the definitions of Caputo derivatives, since they can be applied to a larger set of functions than (A.32)-(A.33).

Whatever the definition employed,

R

I 0 f ( x )= D 0 f ( x )= f ( x ) .

R

RL D m

L D m

c D m

a + f ( x )=

+ f ( x )=

a + f ( x )=

(A.41)

d m f ( x ) d x m

= GL D m

m f ( x )=

a + f ( x )= D

,

RL I m

f ( x )= L D m −

f ( x )= c D m b −

f ( x )=

(A.42)

b −

d m f ( x ) d x m

= GL D m

b − f ( x )=( − 1 ) m D m f ( x )=( − 1 ) m

.

R Some authors do not distinguish the definition employed by means of a superscript (GL, RL, C, L), but use different fonts for the operator instead (D, D , D , D , D ). The particular correspondence between fonts and definitions varies. Very often no indication at all is given, save perhaps in the accompanying text, and the reader is presumed to understand from the context which particular definition is intended.

R In the literature, several alternative notations for operator D may be found: D α a + f ( x )=( D α a + f )( x )= a D α x f ( x )= a I − α x f ( x )= D α x − a f ( x )= d α f ( x ) d ( x − a ) α

(A.43)

d α f ( x ) d ( b − x )

D α

α b −

α b f ( x )= x I − α b

α b − x

f ( x )=( D

f )( x )= x D

f ( x )= D

f ( x )=

(A.44)

α

b −