Grunwald-Letnikov differintegral

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In mathematics, the combined differentiation/integration operator used in fractional calculus is called the differintegral. It takes a few different forms, depending on context. The Grunwald-Letnikov differintegral has one of the simplest definitions, and is a commonly used form of the differintegral. It was introduced by Anton Karl Grünwald (1838-1920) from Prague, in 1867, and by Aleksey Vasilievich Letnikov (1837-1888) in Moscow in 1868.

It is a heuristic extension of the definition of the derivative:

f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}.

[edit] Constructing the Grunwald-Letnikov differintegral

The formula for the derivative can be applied recursively to get higher-order derivatives. For example, the second-order derivative would be:

f''(x) = \lim_{h \to 0} \frac{f'(x+h)-f'(x)}{h}
= \lim_{h_1 \to 0} \frac{\lim_{h_2 \to 0} \frac{f(x+h_1+h_2)-f(x+h_1)}{h_2}-\lim_{h_2 \to 0} \frac{f(x+h_2)-f(x)}{h_2}}{h_1}

Assuming that the h 's converge synchronously, this simplifies to:

= \lim_{h \to 0} \frac{f(x+2h)-2f(x+h)+f(x)}{h^2}

In general, we have (see binomial coefficient):

d^n f(x) = \lim_{h \to 0} \frac{\sum_{0 \le m \le n}(-1)^m {n \choose m}f(x+(n-m)h)}{h^n}

Formally, removing the restriction that n be a positive integer, we have:

\mathbb{D}^q f(x) = \lim_{h \to 0} \frac{1}{h^q}\sum_{0 \le m < \infty}(-1)^m {q \choose m}f(x+(q-m)h)

This defines the Grunwald-Letnikov differintegral.

[edit] Another notation

We may also write the expression more simply if we make the substitution:

\Delta^q_h f(x) = \sum_{0 \le m < \infty}(-1)^m {q \choose m}f(x+(q-m)h)

This results in the expression:

\mathbb{D}^q f(x) = \lim_{h \to 0}\frac{\Delta^q_h f(x)}{h^q}.