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

: $int\_\{a\}^\{b\}\; frac\{f(t)\}\{(t-x)^2\},\; dt\; =\; lim\_\{varepsilon\; o\; 0\}\; left\{\; int\_a^\{x-varepsilon\}frac\{f(t)\}\{(t-x)^2\},dt\; +\; int\_\{x+varepsilon\}^bfrac\{f(t)\}\{(t-x)^2\},dt\; -frac\{2f(x)\}\{varepsilon\}\; ight\}.$