Hadamard finite part integral

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In mathematics, the Hadamard finite part, named after Jacques Hadamard, is a special type of integral for function with hypersingularities.

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 by 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\}.