Hadamard regularization

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 + \int_{x+\varepsilon}^b\frac{f(t)}{(t-x)^2}\,dt -\frac{2f(x)}{\varepsilon}\right\}.