Cauchy Principal Value - Formulation

Formulation

Depending on the type of singularity in the integrand f, the Cauchy principal value is defined as one of the following:

• the finite number

where b is a point at which the behavior of the function f is such that

for any a < b and

for any c > b (one sign is "+" and the other is "−"; see plus or minus for precise usage of notations ±, ∓).

or

• the finite number

where

and

(again, see plus or minus for precise usage of notation ±, ∓).

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

or

• in terms of contour integrals of a complex-valued function f(z); z = x + iy, with a pole on the contour. The pole is enclosed with a circle of radius ε and the portion of the path outside this circle is denoted L(ε). Provided the function f(z) is integrable over L(ε) no matter how small ε becomes, then the Cauchy principal value is the limit:

where two of the common notations for the Cauchy principal value appear on the left of this equation.

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.

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