In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Grünwald–Letnikov derivative is a basic extension of the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 in fractional calculus

Fractional calculus

Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.and the integration operator J...

, that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague

Prague

Prague is the capital and largest city of the Czech Republic. Situated in the north-west of the country on the Vltava river, the city is home to about 1.3 million people, while its metropolitan area is estimated to have a population of over 2.3 million...

, in 1867, and by Aleksey Vasilievich Letnikov

Aleksey Letnikov

Aleksey Vasilievich Letnikov , was a Russian mathematician.Aleksey graduated from the Constantin Institute for Border-lines in Moscow. After graduation he attended classes at the Moscow University and Sorbonne. In 1860 he became an Instructor of Mathematics at the Constantin Institute. He...

 (1837-1888) in Moscow

Moscow

Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...

 in 1868.

Constructing the Grünwald–Letnikov derivative

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



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


which can be justified rigorously by the mean value theorem

Mean value theorem

In calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...

. In general, we have (see binomial coefficient

Binomial coefficient

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

):


Formally, removing the restriction that n be a positive integer, it is reasonable to define:


This defines the Grünwald–Letnikov derivative.

Another notation

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


This results in the expression: