Fractional Integral

Author

Eric W. Weisstein
April 14, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/FractionalIntegral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FractionalIntegral.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Code

FractionalD[nu_,f_,t_,opts___]:=Integrate[(t-x)^(-nu-1)(f/.t->x),{x,0,t},opts]/Gamma[-nu]

FractionalD[mu_?Positive,f_,t_,opts___]:=Module[{m=Ceiling[mu]},
D[FractionalD[-(m-mu),f,t,opts],{t,m}]
]

Power

FractionalD[-ν,t^λ,t,Assumptions->{t>0&&Re[λ]>-1&&Re[ν]>0}]

Constant Function

Limit[(((t^(λ+ν)) Gamma[1+λ])/(Gamma[1+λ+ν])),λ->0]

Exp

FullSimplify[FractionalD[-ν,Exp[a t],t,Assumptions->{t>0,Re[ν]>0,Re[a]>0}]]

Integrate[x^(ν-1) Exp[-a x],{x,0,t},Assumptions->t>0&&Re[a]>0&&Re[ν]>0]

FullSimplify[Gamma[ν,0,a t]-(Gamma[ν]-Gamma[ν,a t])]

Simplify[D[Gamma[λ+1]/Gamma[λ+m-μ+1] t^(λ+m-μ),t]]

Simplify[D[Gamma[λ+1]/Gamma[λ+m-μ+1] t^(λ+m-μ),{t,2}]]

Simplify[D[Gamma[λ+1]/Gamma[λ+m-μ+1] t^(λ+m-μ),{t,3}]]

FullSimplify[Product[n+λ-μ,{n,m}]]

FullSimplify[Gamma[λ+1]Pochhammer[1+λ-μ,m] t^(λ-μ)/Gamma[1+m+λ-μ]]

Semi-Integral

(((t^(λ+ν)) Gamma[1+λ])/(Gamma[1+λ+ν]))/.ν->1/2

Limit[%,λ->0]