structural:summarization
Formulation

Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules:

with a < b < c {\displaystyle a<b<c} and where b is the difficult point, at which the behavior of the function f is such that

(See plus or minus for the precise use of notations ± and ∓.)

In some cases it is necessary to deal simultaneously with singularities both at a finite number b and at infinity. This is usually done by a limit of the form

In those cases where the integral may be split into two independent, finite limits,

and

then the function is integrable in the ordinary sense. The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, it is technically not a "principal value".

The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function f (z): z = x + i y, {\displaystyle f(z):z=x+i\,y\;,} with x, y ∈ R, {\displaystyle x,y\in \mathbb {R} \;,} with a pole on a contour C. Define C (ε) {\displaystyle C(\varepsilon)} to be that same contour, where the portion inside the disk of radius ε around the pole has been removed. Provided the function f (z) {\displaystyle f(z)} is integrable over C (ε) {\displaystyle C(\varepsilon)} no matter how small ε becomes, then the Cauchy principal value is the limit:

In the case of Lebesgue-integrable functions, that is, functions which are integrable in absolute value, these definitions coincide with the standard definition of the integral.

If the function f (z) {\displaystyle f(z)} is meromorphic, the Sokhotski–Plemelj theorem relates the principal value of the integral over C with the mean-value of the integrals with the contour displaced slightly above and below, so that the residue theorem can be applied to those integrals.

Principal value integrals play a central role in the discussion of Hilbert transforms.