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. Formulation. Depending on the type of singularity in the integrand , 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 and at infinity. This is usually done by a limit of the form formula_1 In those cases where the integral may be split into two independent, finite limits, formula_2 and formula_3 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 formula_4 with formula_5 with a pole on a contour . Define formula_6 to be that same contour, where the portion inside the disk of radius around the pole has been removed. Provided the function formula_7 is integrable over formula_6 no matter how small becomes, then the Cauchy principal  value is the limit: formula_9 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 formula_7 is "meromorphic", the Sokhotski–Plemelj theorem relates the principal value of the integral over 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. Distribution theory. Let formula_11 be the set of bump functions, i.e., the  space of smooth functions with compact support on the real line formula_12. Then the map formula_13 defined via the Cauchy principal value as formula_14 is a distribution. The map itself may sometimes be called the principal value (hence the notation p.v.). This distribution appears, for example, in the Fourier transform of the Sign function and the Heaviside step function. Well-definedness as a distribution. To prove the existence of the limit formula_15 for a Schwartz function formula_16, first observe that formula_17 is continuous on formula_18 as formula_19 and hence formula_20 since formula_21 is continuous and L'Hopital's rule applies. Therefore , formula_22 exists and by applying the mean value theorem to formula_23 we get: And furthermore: we note that the map formula_26 is bounded by the usual seminorms for Schwartz functions formula_27. Therefore, this map defines, as it is obviously linear, a continuous functional on the Schwartz space and therefore a tempered distribution. Note that the proof needs formula_28 merely to be continuously differentiable in a neighbourhood of 0 and formula_29 to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as formula_28 integrable with compact support and differentiable at 0. More general definitions. The  principal value is the inverse distribution of the function formula_31 and is almost the only distribution with this property: formula_32 where formula_33 is a constant and formula_34 the Dirac distribution. In a broader sense, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space formula_35. If formula_33 has an isolated singularity at the origin, but is an otherwise "nice" function, then the principal-value distribution is defined on compactly supported smooth functions by formula_37 Such a limit may not be well defined, or, being well-defined, it may not necessarily define a  distribution. It is, however, well-defined if formula_33 is a continuous homogeneous function of degree formula_39 whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms. Examples. Consider the values of two limits: formula_40 This is the Cauchy principal value of the otherwise ill-defined expression formula_41 Also: formula_42 Similarly, we have formula_43 This is the principal value of the otherwise ill-defined expression formula_44 but formula_45 Notation. Different authors use different notations for the Cauchy principal value of a function formula_46, among others: formula_47 formula_48 formula_49 formula_50 as well  as formula_51 P.V., formula_52 formula_53 formula_54 and V.P.