Hadamard regularization

Renormalization and regularization
Renormalization Renormalization Renormalization group
Regularization Dimensional regularization Pauli–Villars regularization Lattice regularization Zeta function regularization Causal perturbation theory Hadamard regularization Point-splitting regularization

In mathematics, Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part, introduced by Hadamard (1923, book III, chapter I, 1932). Riesz (1938, 1949) showed that this can be interpreted as taking the meromorphic continuation of a convergent integral.

If the Cauchy principal value integral

$\int_a^b \frac{f(t)}{t-x} \, dt$

exists, then the Hadamard finite part integral can be defined as

$\int_a^b \frac{f(t)}{(t-x)^2}\, dt = \frac{d}{dx} \int_{a}^{b} \frac{f(t)}{t-x} \,dt.$

Also it can be calculated from the definition

$\int_a^b \frac{f(t)}{(t-x)^2}\, dt = \lim_{\varepsilon \to 0} \left\{ \int_a^{x-\varepsilon}\frac{f(t)}{(t-x)^2}\,dt %2B \int_{x%2B\varepsilon}^b\frac{f(t)}{(t-x)^2}\,dt -\frac{2f(x)}{\varepsilon}\right\}.$

References

Blanchet, Luc; Faye, Guillaume (2000), "Hadamard regularization", Journal of Mathematical Physics 41 (11): 7675–7714, doi:10.1063/1.1308506, ISSN 0022-2488, MR 1788597
Hadamard, Jacques (1923), Lectures on Cauchy's problem in linear partial differential equations, Dover Phoenix editions, Dover Publications, New York, ISBN 978-0-486-49549-1, JFM 49.0725.04, MR 0051411, http://books.google.com/books?id=B25O-x21uqkC
Hadamard, J. (1932) (in French), Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques, Paris: Hermann & Cie., pp. 542, Zbl 0006.20501
Riesz, Marcel (1938), "Intégrales de Riemann-Liouville et potentiels." (in French), Acta Litt. Sci. Szeged 9: 1–42, Zbl 0018.40704
Riesz, Marcel (1949), "L'intégrale de Riemann-Liouville et le problème de Cauchy", Acta Mathematica 81: 1–223, doi:10.1007/BF02395016, ISSN 0001-5962, MR 0030102