Cauchy distribution in the left side of the integral is: where . Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written: The series is just the Taylor expansion for the logarithm of so the entropy may be written in closed form as: Circular Cauchy distribution If X is Cauchy distributed with median μ and scale parameter γ, then the complex variable has unit modulus and is distributed on the unit circle with density: where and ψ expresses the two parameters of the associated linear Cauchy distribution for x as a complex number: It can be seen that the circular Cauchy distribution has the same functional form as the wrapped Cauchy distribution in z and ζ (i.e. fWC(z,ζ)). The circular Cauchy distribution is a reparameterized wrapped Cauchy distribution: The distribution is called the circular Cauchy distribution (also the complex Cauchy distribution) with parameters μ and γ. (See also McCullagh's parametrization of the Cauchy distributions and Poisson kernel for related concepts.) The circular Cauchy distribution expressed in complex form has finite moments of all orders for integer n ≥ 1. For |φ| < 1, the transformation is holomorphic on the unit disk, and the transformed variable U(Z, φ) is distributed as complex Cauchy with parameter U(ζ, φ). Given a sample z1, ..., zn of size n > 2, the maximum-likelihood equation can be solved by a simple fixed-point iteration: starting with ζ(0) = 0. The sequence of likelihood values is non-decreasing, and the solution is unique for samples containing at least three distinct values. The maximum-likelihood estimate for the median () and scale parameter () of a real Cauchy sample is obtained by the inverse transformation: For n ≤ 4, closed-form expressions are known for . The density of the maximum-likelihood estimator at t in the unit disk is necessarily of the form: where . Formulae for p3 and p4 are available. See also Wrapped distribution Dirac comb Wrapped normal distribution Circular uniform distribution McCullagh's parametrization of the Cauchy distributions References Continuous distributions Directional statistics