## Fractional Calculus

Denote the th Derivative and the -fold Integral . Then

 (1)

Now, if
 (2)

is true for , then
 (3)

Interchanging the order of integration gives
 (4)

But (2) is true for , so it is also true for all by Induction. The fractional integral of can then be defined by
 (5)

where is the Gamma Function.

The fractional integral can only be given in terms of elementary functions for a small number of functions. For example,

 (6) (7)

where is the Et-Function. The fractional derivative of (if it exists) can be defined by
 (8)

An example is
 (9) (10)

It is always true that, for ,
 (11)

but not always true that
 (12)

References

Love, E. R. Fractional Derivatives of Imaginary Order.'' J. London Math. Soc. 3, 241-259, 1971.

McBride, A. C. Fractional Calculus. New York: Halsted Press, 1986.

Miller, K. S. Derivatives of Noninteger Order.'' Math. Mag. 68, 183-192, 1995.

Nishimoto, K. Fractional Calculus. New Haven, CT: University of New Haven Press, 1989.

Spanier, J. and Oldham, K. B. The Fractional Calculus. New York: Academic Press, 1974.