Hadamard finite part integral

Hadamard finite part integral

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}^bfrac{f(t)}{(t-x)^2},dt -frac{2f(x)}{varepsilon}right}.

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