Cauchy principal value

In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules: 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 The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function f(z); z = x + iy, with a pole on a contour C. Define C(ε) to be the same contour where the portion inside the disk of radius ε around the pole has been removed. Provided the function f(z) is integrable over C(ε) 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) 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. Let C c ∞ ( R ) {displaystyle {C_{c}^{infty }}(mathbb {R} )} be the set of bump functions, i.e., the space of smooth functions with compact support on the real line R {displaystyle mathbb {R} } . Then the map