mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

fractional calculus Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D D f(x) = \frac f(x)\,, and of the integration ...

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 ( ; ) is the capital and List of cities and towns in the Czech Republic, largest city of the Czech Republic and the historical capital of Bohemia. Prague, located on the Vltava River, has a population of about 1.4 million, while its P ...

, in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in

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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: :

\beginf''(x)&=\lim_\frac\\&=\lim_\frac\end

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

= \lim_ \frac

which can be justified rigorously by the

mean value theorem In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...

. In general, we have (see

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

): :

f^(x) = \lim_ \frac

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).

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

\mathbb^q f(x) =  \lim_\frac.

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 :

f^(x) = \lim_ \frac\sum_(-1)^m f(x+mh).

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

\mathbb^q f(x) = \lim_ \frac\sum_(-1)^m f(x+mh).

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: :

\mathbb^q f(x) = \lim_ \frac\sum_(-1)^m f(x-mh).

Constructing the Grünwald–Letnikov derivative

An alternative definition

References

Further reading