Editing Pi (section)

=== Cauchy distribution and potential theory ===
[[File:Witch of Agnesi, construction.svg|thumb|The [[Witch of Agnesi]], named for [[Maria Gaetana Agnesi|Maria Agnesi]] (1718–1799), is a geometrical construction of the graph of the Cauchy distribution.|left]]
[[File:2d random walk ag adatom ag111.gif|left|thumb|The Cauchy distribution governs the passage of [[Brownian motion|Brownian particles]] through a membrane.]]
The [[Cauchy distribution]]

<math display=block>g(x)=\frac{1}{\pi}\cdot\frac{1}{x^2+1}</math>

is a [[probability density function]].  The total probability is equal to one, owing to the integral:

<math display=block>\int_{-\infty }^{\infty } \frac{1}{x^2+1} \, dx = \pi.</math>

The [[Shannon entropy]] of the Cauchy distribution is equal to {{math|ln(4π)}}, which also involves {{pi}}.

The Cauchy distribution plays an important role in [[potential theory]] because it is the simplest [[Furstenberg boundary|Furstenberg measure]], the classical [[Poisson kernel]] associated with a [[Brownian motion]] in a half-plane.<ref>{{cite book |first1=Sidney |last1=Port |first2=Charles |last2=Stone |title=Brownian motion and classical potential theory |publisher=Academic Press |year=1978 |page=29}}</ref> [[Conjugate harmonic function]]s and so also the [[Hilbert transform]] are associated with the asymptotics of the Poisson kernel. The Hilbert transform ''H'' is the integral transform given by the [[Cauchy principal value]] of the [[singular integral]]

<math display=block>Hf(t) = \frac{1}{\pi}\int_{-\infty}^\infty \frac{f(x)\,dx}{x-t}.</math>

The constant {{pi}} is the unique (positive) normalizing factor such that ''H'' defines a [[linear complex structure]] on the Hilbert space of square-integrable real-valued functions on the real line.<ref>{{cite book |last=Titchmarsh |first=E. |author-link=Edward Charles Titchmarsh |title=Introduction to the Theory of Fourier Integrals |isbn=978-0-8284-0324-5 |year=1948 |edition=2nd |publication-date=1986 |publisher=Clarendon Press |location=Oxford University}}</ref>  The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space {{math|L<sup>2</sup>('''R''')}}: up to a normalization factor, it is the unique bounded linear operator that commutes with positive dilations and anti-commutes with all reflections of the real line.<ref>{{cite book |first=Elias |last=Stein |author-link=Elias Stein |title=Singular Integrals and Differentiability Properties of Functions |publisher=Princeton University Press |year=1970}}; Chapter II.</ref>  The constant {{pi}} is the unique normalizing factor that makes this transformation unitary.