finite part | partie finie | principal value integral

The Cauchy principal value of a finite integral of a function f about a point c with a<=c<=b is given by P V integral_a^b f(x) d x congruent lim_(ϵ->0^+)[ integral_a^(c - ϵ) f(x) d x + integral_(c + ϵ)^b f(x) d x] (Henrici 1988, p. 261, p. 117, p. 283). Similarly, the Cauchy principal value of a doubly infinite integral of a function f is defined by P V integral_(-∞)^∞ f(x) d x congruent lim_(R->∞) integral_(-R)^R f(x) d x. The Cauchy principal value is also known as the principal value integral, finite part, or partie finie.

Hilbert transform | logarithmic integral | principal value | simple pole