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Reverted 1 good faith edit by 86.165.176.69 using STiki

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	~~{{Refimprove\|date=June 2009}}~~		Man or woman who wrote the article is called Roberto Ledbetter and his wife a lot like it at every bit. In his professional life he can be a people manager. He's always loved living for Guam and he features everything that he could use there. The [http://Www.Reddit.com/r/howto/search?q=preference+hobby preference hobby] for him and simply his kids is landscaping but he's been claiming on new things not too lengthy ago. He's been working on this website for some a chance now. Check it completly here: http://circuspartypanama.com<br><br>
	~~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 Letnikov\|Aleksey Vasilievich Letnikov]] (1837–1888) in [[Moscow]] in 1868~~.

	~~==Constructing the Grünwald–Letnikov derivative==~~		Have a look at my site ... [http://circuspartypanama.com Clash Of Clans Hack Android]

	~~The formula~~
	~~:<math>f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}</math>~~
	~~for the derivative can be applied recursively to get higher-order derivatives. For example, the second-order derivative would be:~~

	~~:<math>f''(x) = \lim_{h \to 0} \frac{f'(x+h)-f'(x)}{h}</math>~~

	~~:<math> = \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}</math>~~

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

	~~:<math> = \lim_{h \to 0} \frac{f(x+2h)-2f(x+h)+f(x)}{h^2},</math>~~

	~~which can be justified rigorously by the [[mean value theorem]]. In general, we have (see [[binomial coefficient]]):~~

	~~:<math>f^{(n)}(x) = \lim_{h \to 0} \frac{\sum_{0 \le m \le n}(-1)^m {n \choose m}f(x+(n-m)h)}{h^n}.</math>~~

	~~Removing the restriction that ''n'' be~~ a ~~positive integer, it is reasonable to define:~~

	~~:<math>\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)~~.~~</math>~~

	~~This defines the Grünwald–Letnikov derivative~~.

	~~To simplify notation, we set:~~

	~~:<math>\Delta^q_h f(x) = \sum_{0 \le m < \infty}(-1)^m {q \choose m}f(x+(q-m)h)~~.~~</math>~~

	~~So the Grünwald–Letnikov derivative may be succinctly written as:~~

	~~:<math>\mathbb{D}^q f(x) = \lim_{h \to 0}\frac{\Delta^q_h f(x)}{h^q}.</math>~~

	~~===An alternative definition===~~

	~~In the preceding section, the general first principles equation for integer order derivatives was derived. It can be shown that the equation may also be written as~~

	~~:<math>f^{(n)}(x) = \lim_{h \to 0} \frac{(-1)^n}{h^n}\sum_{0 \le m \le n}(-1)^m {n \choose m}f(x+mh).</math>~~

	~~or removing the restriction that ''n'' must be a positive integer:~~

	~~:<math>\mathbb{D}^q f(x) = \lim_{h \to 0} \frac{(-1)^q}{h^q}\sum_{0 \le m < \infty}(-1)^m {q \choose m}f(x+mh).</math>~~

	~~This equation is called the reverse Grünwald–Letnikov derivative. If the substitution ''h'' → −''h'' is made, the resulting equation is called the direct Grünwald–Letnikov derivative:<ref>~~http://~~www.diogenes.bg/fcaa/volume7/fcaa74/74_Ortigueira_Coito~~.~~pdf</ref>~~

	~~:<math>\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-mh).</math>~~

	~~== References ==~~

	~~<references />~~

	* ''The Fractional Calculus'', by Oldham, K.; and Spanier, J. Hardcover: 234 pages. Publisher: Academic Press, 1974. ISBN 0-12-525550-0
	* ''From Differences to Derivatives'', by Ortigueira, M. D., and F. Coito. Fractional Calculus and Applied Analysis 7(4). (2004): 459-71.

	~~{{DEFAULTSORT:Grunwald-Letnikov Derivative}}~~
	~~[[Category:Fractional calculus]~~]

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← Older revision		Revision as of 16:34, 25 November 2013
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			{{Refimprove\|date=June 2009}}
			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 Letnikov\|Aleksey Vasilievich Letnikov]] (1837–1888) in [[Moscow]] in 1868.

			==Constructing the Grünwald–Letnikov derivative==

	~~Roberto is what~~'s ~~written concerning his birth certificate rather he never really adored that name~~. ~~Managing people is where his primary financial comes from~~. ~~Base jumping is something~~ that ~~bigger been doing for~~ a ~~number of~~. ~~Massachusetts has always been his living place and his wife and kids loves it~~. ~~Go in which~~ to ~~his website~~ to ~~obtain out more~~: ~~http~~://[http://~~Www~~.~~Guardian~~.Co.uk/~~search?~~q=~~prometeu~~.~~net prometeu.net]~~<br><br>~~Have a look at my web page~~ - ~~hack clash of clans~~ ([~~http~~:~~//prometeu.net try these out~~])		The formula
			:<math>f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}</math>
			for the derivative can be applied recursively to get higher-order derivatives. For example, the second-order derivative would be:

			:<math>f''(x) = \lim_{h \to 0} \frac{f'(x+h)-f'(x)}{h}</math>

			:<math> = \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}</math>

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

			:<math> = \lim_{h \to 0} \frac{f(x+2h)-2f(x+h)+f(x)}{h^2},</math>

			which can be justified rigorously by the [[mean value theorem]]. In general, we have (see [[binomial coefficient]]):

			:<math>f^{(n)}(x) = \lim_{h \to 0} \frac{\sum_{0 \le m \le n}(-1)^m {n \choose m}f(x+(n-m)h)}{h^n}.</math>

			Removing the restriction that ''n'' be a positive integer, it is reasonable to define:

			:<math>\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).</math>

			This defines the Grünwald–Letnikov derivative.

			To simplify notation, we set:

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

			So the Grünwald–Letnikov derivative may be succinctly written as:

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

			===An alternative definition===

			In the preceding section, the general first principles equation for integer order derivatives was derived. It can be shown that the equation may also be written as

			:<math>f^{(n)}(x) = \lim_{h \to 0} \frac{(-1)^n}{h^n}\sum_{0 \le m \le n}(-1)^m {n \choose m}f(x+mh).</math>

			or removing the restriction that ''n'' must be a positive integer:

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

			This equation is called the reverse Grünwald–Letnikov derivative. If the substitution ''h'' → −''h'' is made, the resulting equation is called the direct Grünwald–Letnikov derivative:<ref>http://www.diogenes.bg/fcaa/volume7/fcaa74/74_Ortigueira_Coito.pdf</ref>

			:<math>\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-mh).</math>

			== References ==

			<references />

			* ''The Fractional Calculus'', by Oldham, K.; and Spanier, J. Hardcover: 234 pages. Publisher: Academic Press, 1974. ISBN 0-12-525550-0
			* ''From Differences to Derivatives'', by Ortigueira, M. D., and F. Coito. Fractional Calculus and Applied Analysis 7(4). (2004): 459-71.

			{{DEFAULTSORT:Grunwald-Letnikov Derivative}}
			[[Category:Fractional calculus]]

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Roberto is what's written concerning his birth certificate rather he never really adored that name. Managing people is where his primary financial comes from. Base jumping is something that bigger been doing for a number of. Massachusetts has always been his living place and his wife and kids loves it. Go in which to his website to obtain out more: http://[http://Www.Guardian.Co.uk/search?q=prometeu.net prometeu.net]<br><br>Have a look at my web page - hack clash of clans ([http://prometeu.net try these out])