Editing Cauchy principal value (section)

===More general definitions===

The principal value is the inverse distribution of the function <math> x </math> and is almost the only distribution with this property:
<math display="block"> x f = 1 \quad \Leftrightarrow \quad \exists K: \; \; f = \operatorname{p.\!v.} \left( \frac{1}{x} \right) + K \delta, </math>
where <math> K </math> is a constant and <math> \delta </math> the Dirac distribution.

In a broader sense, the principal value can be defined for a wide class of [[singular integral]] [[integral kernel|kernels]] on the Euclidean space <math> \mathbb{R}^{n} </math>. If <math> K </math> 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
<math display="block"> [\operatorname{p.\!v.} (K)](f) = \lim_{\varepsilon \to 0} \int_{\mathbb{R}^{n} \setminus B_{\varepsilon}(0)} f(x) K(x) \, \mathrm{d} x. </math>
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 <math> K </math> is a continuous [[homogeneous function]] of degree <math> -n </math> whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the [[Riesz transform]]s.