Definitions

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

$int_\left\{a\right\}^\left\{b\right\} frac\left\{f\left(t\right)\right\}\left\{t-x\right\} , dt$

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

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

Also it can be calculated by definition

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

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