The Cauchy principal value of a finite integral of a function about a point with is given by
(Henrici 1988, p. 261; Whittaker and Watson 1990, p. 117; Bronshtein and Semendyayev 1997, p. 283). Similarly, the Cauchy principal value of a doubly
infinite integral of a function is defined by
The Cauchy principal value is also known as the principal value integral (Henrici 1988, p. 261), finite part (Vladimirov 1971), or partie finie (Vladimirov 1971).
The Cauchy principal value of an integral having no nonsimple poles can be computed in the Wolfram Language using Integrate[f,
x,
a, b,
PrincipalValue -> True]. Cauchy principal values of functions with possibly
nonsimple poles can be computed numerically using the "CauchyPrincipalValue"
method in NIntegrate.
Cauchy principal values are important in the theory of generalized functions, where they allow extension of results to .
Cauchy principal values are sometimes simply known as "principal values" (e.g., Vladimirov 1971, p. 75) even though they are not related to the principal
value of complex analysis.
The most common designation for the Cauchy principal values seems to be (Henrici 1988, pp. 259-262; Gradshteyn and
Ryzhik 2000, p. 523). Sometimes, no explicit designation is used (Harris and
Stocker 1998, p. 552; Gradshteyn and Ryzhik 2000, p. 248). Other notations
include
(Arfken 1985, p. 403), P.V. (Apelblat 1983, p. viii), (Morse and Feshbach 1953, p. 368; most Russian authors),
(Vladimirov 1971), (Bronshtein and Semendyayev 1997, p. 282), and V.P.
(Brychkov 1992, p. 7). For integrals with finite limits, the Cauchy principal
value is sometimes denoted (Zwillinger 1995, p. 346).