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Grunwald–Letnikov differintegral : ウィキペディア英語版
Grünwald–Letnikov derivative

In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague, in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 1868.
==Constructing the Grünwald–Letnikov derivative==

The formula
:f'(x) = \lim_ \frac
for the derivative can be applied recursively to get higher-order derivatives. For example, the second-order derivative would be:
:f''(x) = \lim_ \frac
: = \lim_ \frac-\lim_ \frac}
Assuming that the ''h'' 's converge synchronously, this simplifies to:
: = \lim_ \frac,
which can be justified rigorously by the mean value theorem. In general, we have (see binomial coefficient):
:f^(x) = \lim_ \fracf(x+(n-m)h)}.
Removing the restriction that ''n'' be a positive integer, it is reasonable to define:
:\mathbb^q f(x) = \lim_ \frac\sum_(-1)^m f(x+(q-m)h).
This defines the Grünwald–Letnikov derivative.
To simplify notation, we set:
:\Delta^q_h f(x) = \sum_(-1)^m f(x+(q-m)h).
So the Grünwald–Letnikov derivative may be succinctly written as:
:\mathbb^q f(x) = \lim_\frac.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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