Denote the th derivative and the fold integral . Then
(1)

Now, if the equation
(2)

for the multiple integral is true for , then
(3)
 
(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 RiemannLiouville 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 semiintegral.
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 Etfunction. 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.