Cauchy principal value (of an integral)

P.V. $\int_{-\infty}^\infty f(x)\,dx := \lim_{R\to\infty}\int_{-R}^R f(x)\,dx$ , a symmetric limit that can exist even when the ordinary improper integral does not.

The Cauchy principal value of $\int_{-\infty}^\infty f(x)\,dx$ , written P.V. $\int_{-\infty}^\infty f(x)\,dx$ , is the symmetric limit $\lim_{R\to\infty}\int_{-R}^R f(x)\,dx$ ; when $f$ has a singularity on the real axis at $x_0$ , the same idea applies by symmetrically excluding a shrinking interval $(x_0-\varepsilon,x_0+\varepsilon)$ before taking $\varepsilon\to 0$ . This symmetric limit can exist and be finite even when the two-sided ordinary improper integral (letting each end run off independently) does not, which is why residue calculus computations are always understood as computing the principal value unless stated otherwise.

Examples

P.V. $\int_{-\infty}^\infty x\,dx=0$ by symmetry, even though $\int_{-\infty}^\infty x\,dx$ does not converge as an ordinary improper integral.
Indenting a contour around a pole on the real axis and letting the indentation shrink computes exactly the Cauchy principal value of the corresponding real integral.