“Fractional Calculus”的概念、定义、习题解题方法及翻译-数学辞典

单词

Fractional Calculus

释义

Fractional Calculus

Denote the th Derivative and the -fold Integral . Then

(1)

Now, if

(2)

is true for

, then

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. Forexample,

			(6)
			(7)

where

is the Et-Function. The fractional derivative of

(if it exists) can be defined by

(8)

An example is

It is always true that, for

(11)

but not always true that

(12)

See also Derivative, Integral

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.

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