Cauchy Principal Value

$PV \int^\infty_{-\infty} f(x)\,dx \equiv \lim_{R\to \infty} \int^R_{-R} f(x)\,dx$

$PV \int^b_a f(x)\,dx \equiv\lim_{\epsilon\to 0} \left[{\int^{c-\epsilon}_a f(x)\,dx+\int^b_{c+\epsilon}f(x)\,dx}\right],$

where $\epsilon > 0$ and $a\leq c\leq b$ .

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 401-403, 1985.

Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 158, 1991.

$PV \int^\infty_{-\infty} f(x)\,dx \equiv \lim_{R\to \infty} \int^R_{-R} f(x)\,dx$
$PV \int^b_a f(x)\,dx \equiv\lim_{\epsilon\to 0} \left[{\int^{c-\epsilon}_a f(x)\,dx+\int^b_{c+\epsilon}f(x)\,dx}\right],$