Denote the th Derivative and the -fold Integral . Then

(1) |

(2) |

(3) |

Interchanging the order of integration gives

(4) |

(5) |

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) |

(9) | |||

(10) |

It is always true that, for ,

(11) |

(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.

© 1996-9

1999-05-26