Denote the th derivative and the -fold integral . Then
(1)
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Now, if the equation
(2)
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for the multiple integral is true for , then
(3)
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(4)
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Interchanging the order of integration gives
(5)
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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)
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where is the gamma function.
More generally, the Riemann-Liouville operator of fractional integration is defined as
(7)
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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)
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(9)
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(10)
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(11)
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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)
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(13)
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A fractional derivative can also be similarly defined. The study of fractional derivatives and integrals is called fractional calculus.