section_path: Role and characterizations in mathematics/Cauchy distribution and potential theory
section_name: Cauchy distribution and potential theory
paragraph_text: Cauchy distribution and potential theory
  
  The Cauchy distribution g (x) = 1 π ⋅ 1 x 2 + 1 {\displaystyle g(x)={\frac {1}{\pi }}\cdot {\frac {1}{x^{2}+1}}} is a probability density function. The total probability is equal to one, owing to the integral: ∫ − ∞ ∞ 1 x 2 + 1 d x = π. {\displaystyle \int _{-\infty }^{\infty }{\frac {1}{x^{2}+1}}\,dx=\pi.}
  
  The Shannon entropy of the Cauchy distribution is equal to ln(4π), which also involves π.
  
  The Cauchy distribution plays an important role in potential theory because it is the simplest Furstenberg measure, the classical Poisson kernel associated with a Brownian motion in a half-plane. Conjugate harmonic functions 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 H f (t) = 1 π ∫ − ∞ ∞ f (x) d x x − t. {\displaystyle Hf(t)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {f(x)\,dx}{x-t}}.}
  
  The constant π 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. The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space L (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. The constant π is the unique normalizing factor that makes this transformation unitary.