Weyl integral

Part of a series of articles about

Definitions
Derivative (generalizations) Differential infinitesimal of a function total
Concepts
Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem
Rules and identities
Sum Product Chain Power Quotient General Leibniz Faà di Bruno's formula

Integral

Definitions
Lists of integrals
Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration
Integration by
Parts Discs Cylindrical shells Substitution (trigonometric) Partial fractions Order Reduction formulae

Series

Convergence tests
Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor
Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel

Vector

Theorems
Gradient Divergence Curl Laplacian Directional derivative Identities
Divergence Gradient Green's Kelvin–Stokes

Multivariable

Formalisms
Matrix Tensor Exterior Geometric
Definitions
Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian matrix

Specialized

In mathematics, the Weyl integral 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

\sum_{n=-\infty}^{\infty} a_n e^{in \theta}

with a₀ = 0.

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

\sum_{n=-\infty}^{\infty} (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 a₀ = 0 here plays the obvious role of excluding the need to consider division by zero. The definition is due to Hermann Weyl (1917).

References

Lizorkin, P.I. (2001), "Fractional integration and differentiation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

This article is issued from Wikipedia - version of the 3/17/2013. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Weyl integral

See also

References