Denote the 
th
derivative 
and the 
-fold integral 
. Then
 |
(1)
|
Now, if the equation
 |
(2)
|
for the multiple integral is true for 
, then
 |  | ![D^(-1)[1/((n-1)!)int_0^t(t-xi)^(n-1)f(xi)dxi]](/images/equations/FractionalIntegral/Inline8.svg) |
(3)
|
 |  | ![int_0^t[1/((n-1)!)int_0^x(x-xi)^(n-1)f(xi)dxi]dx.](/images/equations/FractionalIntegral/Inline11.svg) |
(4)
|
Interchanging the order of integration gives
 |
(5)
|
But (3) is true for 
, so it is also true for all 
by induction. The fractional
integral of 
of order 
can then be defined by
 |
(6)
|
where 
is the gamma function.
More generally, the Riemann-Liouville operator of fractional integration is defined as
 |
(7)
|
for 
with 
(Oldham and Spanier 1974, Miller and Ross 1993, Srivastava and Saxena 2001, Saxena
2002).
The fractional integral of order 1/2 is called a semi-integral.
Few functions have a fractional integral expressible in terms of elementary functions. Exceptions include
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
where 
is a lower incomplete gamma function
and 
is the Et-function. From (10),
the fractional integral of the constant function 
is given by
 |  |  |
(12)
|
 |  |  |
(13)
|
A fractional derivative can also be similarly defined. The study of fractional derivatives and integrals is called fractional calculus.
