Editing Cauchy principal value (section)

== Distribution theory ==

Let <math> {C_{c}^{\infty}}(\mathbb{R}) </math> be the set of [[bump function]]s, i.e., the space of [[smooth function]]s with [[compact support]] on the [[real number|real line]] <math> \mathbb{R} </math>. Then the map
<math display="block"> \operatorname{p.\!v.} \left( \frac{1}{x} \right) \,:\, {C_{c}^{\infty}}(\mathbb{R}) \to \mathbb{C} </math>
defined via the Cauchy principal value as
<math display="block"> \left[ \operatorname{p.\!v.} \left( \frac{1}{x} \right) \right](u) = \lim_{\varepsilon \to 0^{+}} \int_{\mathbb{R} \setminus [- \varepsilon,\varepsilon]} \frac{u(x)}{x} \, \mathrm{d} x = \lim_{\varepsilon \to 0^{+}} \int_{\varepsilon}^{+ \infty} \frac{u(x) - u(- x)}{x} \, \mathrm{d} x \quad \text{for } u \in {C_{c}^{\infty}}(\mathbb{R}) </math>
is a [[distribution (mathematics)|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 
<math display="block"> \lim_{\varepsilon \to 0^{+}} \int_{\varepsilon}^{+ \infty} \frac{u(x) - u(- x)}{x} \, \mathrm{d}x </math>
for a [[Schwartz function]] <math>u(x)</math>, first observe that <math>\frac{u(x) - u(-x)}{x}</math> is continuous on <math>[0, \infty),</math> as
<math display="block"> \lim_{\,x \searrow 0\,} \; \Bigl[ u(x) - u(-x) \Bigr] ~= ~0 ~</math>
and hence 
<math display="block"> \lim_{x\searrow 0} \, \frac{u(x) - u(-x)}{x} ~=~ \lim_{\,x\searrow 0\,} \, \frac{u'(x) + u'(-x)}{1} ~=~ 2u'(0)~, </math>
since <math>u'(x)</math> is continuous and [[L'Hopital's rule]] applies.

Therefore, <math>\int_0^1 \, \frac{u(x) - u(-x)}{x} \, \mathrm{d}x</math> exists and by applying the [[mean value theorem]] to <math>u(x) - u(-x) ,</math> we get:
:<math> \left|\, \int_0^1\,\frac{u(x) - u(-x)}{x} \,\mathrm{d}x \,\right|
\;\leq\; \int_0^1 \frac{\bigl|u(x)-u(-x)\bigr|}{x} \,\mathrm{d}x
\;\leq\; \int_0^1\,\frac{\,2x\,}{x}\,\sup_{x \in \mathbb{R} }\,\Bigl|u'(x)\Bigr| \,\mathrm{d}x
\;\leq\; 2\,\sup_{x \in \mathbb{R} }\,\Bigl|u'(x)\Bigr| ~. </math>

And furthermore:
:<math> \left| \,\int_1^\infty \frac {\;u(x) - u(-x)\;}{x} \,\mathrm{d}x \,\right| \;\leq\; 2 \,\sup_{x\in\mathbb{R}} \,\Bigl|x\cdot u(x)\Bigr|~\cdot\;\int_1^\infty \frac{\mathrm{d}x}{\,x^2\,} \;=\; 2 \,\sup_{x\in\mathbb{R}}\, \Bigl|x \cdot u(x)\Bigr| ~, </math>

we note that the map
<math display="block"> \operatorname{p.v.}\;\left( \frac{1}{\,x\,} \right) \,:\, {C_{c}^{\infty}}(\mathbb{R}) \to \mathbb{C} </math>
is bounded by the usual seminorms for [[Schwartz functions]] <math> u</math>. Therefore, this map defines, as it is obviously linear, a continuous functional on the [[Schwartz space]] and therefore a [[distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]].

Note that the proof needs <math>u</math> merely to be continuously differentiable in a neighbourhood of 0 and <math> x\,u </math> to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as <math>u</math> integrable with compact support and differentiable at 0.

===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.