Cauchy Principal Value

• the finite number

$\backslash lim\_\{\backslash varepsilon\backslash rightarrow\; 0+\}\; \backslash left(\backslash int\_a^\{b-\backslash varepsilon\}\; f(x)\backslash ,dx+\backslash int\_\{b+\backslash varepsilon\}^c\; f(x)\backslash ,dx\backslash right)$

where b is a point at which the behavior of the function f is such that

$\backslash int\_a^b\; f(x)\backslash ,dx=\backslash pm\backslash infty$

for any a < b and

$\backslash int\_b^c\; f(x)\backslash ,dx=\backslash mp\backslash infty$

for any c > b (one sign is "+" and the other is "−").

• the finite number

$\backslash lim\_\{a\backslash rightarrow\backslash infty\}\backslash int\_\{-a\}^a\; f(x)\backslash ,dx$

where

$\backslash int\_\{-\backslash infty\}^0\; f(x)\backslash ,dx=\backslash pm\backslash infty$

and

$\backslash int\_0^\backslash infty\; f(x)\backslash ,dx=\backslash mp\backslash infty$

(again, one sign is "+" and the other is "−").

$\backslash lim\_\{\backslash varepsilon\; \backslash rightarrow\; 0+\}\backslash int\_\{b-1/\backslash varepsilon\}^\{b-\backslash varepsilon\}\; f(x)\backslash ,dx+\backslash int\_\{b+\backslash varepsilon\}^\{b+1/\backslash varepsilon\}f(x)\backslash ,dx.$

Nomenclature

Examples

$\backslash lim\_\{a\backslash rightarrow\; 0+\}\backslash left(\backslash int\_\{-1\}^\{-a\}\backslash frac\{dx\}\{x\}+\backslash int\_a^1\backslash frac\{dx\}\{x\}\backslash right)=0,$

$\backslash lim\_\{a\backslash rightarrow\; 0+\}\backslash left(\backslash int\_\{-1\}^\{-a\}\backslash frac\{dx\}\{x\}+\backslash int\_\{2a\}^1\backslash frac\{dx\}\{x\}\backslash right)=-\backslash log\_e\; 2.$

$\backslash int\_\{-1\}^1\backslash frac\{dx\}\{x\}\{\backslash \; \}$

$\backslash lim\_\{a\backslash rightarrow\backslash infty\}\backslash int\_\{-a\}^a\backslash frac\{2x\backslash ,dx\}\{x^2+1\}=0,$

$\backslash lim\_\{a\backslash rightarrow\backslash infty\}\backslash int\_\{-2a\}^a\backslash frac\{2x\backslash ,dx\}\{x^2+1\}=-\backslash log\_e\; 4.$

$\backslash int\_\{-\backslash infty\}^\backslash infty\backslash frac\{2x\backslash ,dx\}\{x^2+1\}\{\backslash \; \}$

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