Tables of Fractional Derivatives

A p p e n d i x : Tables of F>actional D e r i v a t i v e s The short tables below contains Riemann-Liouville fractional derivatives of some flmctions which are fl'equently used in applications. In most cases, the order of differentiation, c~, may be any real number, so replacing it with - a gives the Riemann Liouville fractional integrals. The tables can also be used tbr evaluating the Griinwald Letnikov fractional derivatives, the Caputo fractional dcrivatives, and the MillerRoss sequential fl'actional derivatives as well. In such cases, a, should be taken between 0 and 1, and the Riemann Liouville fractional deriwative should be properly combined with integer- or fractional-order derivatives, in accordance with the considered definition.

1. R i e m a n n - L i o u v i l l e fractional derivatives with the lower terminal at 0

f(t)

oD~*f(t),

(t > O,

a e R)

~mOt

H ( t - a)

H(t - a)f(t)

( t - a) -~

C(1- a)' O,

{ aDp:f(t), O,

(t > o,) (0 < t < a)

(t > a)

(0 < t <

a)

309

310

TABLES OF FRACTIONAL

Illll/Illlllllllllllllll

DERIVATIVES

Illlllllll

f(t)

oD~. f ( t ) ,

(t > O,

ee E R)

t-a-1

a(t)

r(-~) [;--ct--n.-- 1

a(',)(~)

p ( - ~ ± r,,)

('~ ~ N)

(t - a) - n - ~ - I

a(~') (t - ~)

r ( - ~ - ~)

t~

(t > ~)

'

0,

(~ ~ N)

(0 < t < a)

F ( u + 1)

t.+e ~

(u > - 1 )

r ( . + i - ,~)'

e ~t

t-~El,i_~(kt)

co~h(v%)

t-~E2,~_~ (~t 2)

sinh(vFAt)

.t>,~E2 2_~(At 2)

x/~t

in(t)

P(1 - ,~)

+

-

t ~ - i In(t)

r(~)t;'-~-~ r--(a=U (in(t)+ ¢,(:~)- ~(~- ~))

-

(n~.(f~) > o)

(;3>0,

t ~3-a-1E p,,3--a~IAt**~ " ) t

2tq (#, u; fl; At)

r(~

- o:)

2Fi(#,

#>0)

~': fl - a;

(n¢(9) > o) P.y,°(2t-

1)

m ! t -'~ 0,-a p ( m 7- c7-+ 1)~** ( 2 t - 1 ) , (0 < c~ < 1,

0
m=1~2,...

311

T A B L E S OF F R A C T I O N A L D E R I V A T I V E S

2. Riemann-Liouville with

the

lower III

IIIIIIIII

~t(t - a)

terminal

IIIIIIIIl!lll!lll

IIIII

III I

III

I

IIIIIIIIIIIIIIIIIIII

{

derivatives

at -oo

_.~,Dpf(t),

f(t) IIIIIIIIII

fractional

(t > O,

a6R) I

I lUlIIIIIIIIIIIIII

(t-~)-" r(1 - ~)' o,

IIIIIIIIIIIIIIIIII

(t>a) (t < a)

H ( t - a)f(t)

{ ~DTf(t), (t>a)

eat

ArseAt

0,

IIIIIIIIIIIIIIIIIII

(t _< a)

(a > o) (;,At+t*

/~* eAt+/a

(A > o)

sin At

A~sin (At+ gff)

(A>O, a > - l ) cos At

Ac~cos (At + g2q) (A>0,

e At sin #t

rC~e;~t sin (#t + aW)

(,,.=,/~+#2, e At cos #t

0:>-1)

t.a.~=~,

~>o. #>o)

<~e xt cos (#t + a~) (r = ~ / - 2 7 7 1t,2,

tail V) = t~ ,

A>O, # > 0 )