Weyl differintegral

In mathematics, the Weyl differentintegral is an operator defined, as an example of fractional calculus, on functions "f" on the unit circle having integral 0 and a Fourier series. In other words there is a Fourier series for "f" of the form

:Σ "a"_"n""e"^"in"θ

with "a"_"0" = 0, −∞ < "n" < ∞.

Then the Weyl differintegral operator of order "s" is defined on Fourier series by

:&Sigma; ("in")^"s""a"_"n""e"^"in"&theta;

where this is defined. Here "s" can take any real value, and for integer values "k" of "s" the series expansion is the expected "k"-th derivative, if "k" > 0, or −"k"-th indefinite integral normalized by integration from θ = 0.

The condition "a"_"0" = 0 here plays the obvious role of excluding the need to consider division by zero. The definition is due to Hermann Weyl (1917).

External links

* [http://eom.springer.de/f/f041230.htm EoM article Fractional integration and differentiation] ]