In probability theory and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the unit circle. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution. The wrapped Cauchy distribution is often found in the field of spectroscopy where it is used to analyze diffraction patterns (e.g. see Fabry–Pérot interferometer). Description The probability density function of the wrapped Cauchy distribution is: where is the scale factor and is the peak position of the "unwrapped" distribution. Expressing the above pdf in terms of the characteristic function of the Cauchy distribution yields: The PDF may also be expressed in terms of the circular variable z = eiθ and the complex parameter ζ = ei(μ+iγ) where, as shown below, ζ = ⟨z⟩. In terms of the circular variable the circular moments of the wrapped Cauchy distribution are the characteristic function of the Cauchy distribution evaluated at integer arguments: where is some interval of length . The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector: The mean angle is and the length of the mean resultant is yielding a circular variance of 1 − R. Estimation of parameters A series of N measurements drawn from a wrapped Cauchy distribution may be used to estimate certain parameters of the distribution. The average of the series is defined as and its expectation value will be just the first moment: In other words, is an unbiased estimator of the first moment. If we assume that the peak position lies in the interval , then Arg will be a (biased) estimator of the peak position . Viewing the as a set of vectors in the complex plane, the statistic is the length of the averaged vector: and its expectation value is In other words, the statistic will be an unbiased estimator of , and will be a (biased) estimator of . Entropy The information entropy of the wrapped Cauchy distribution is defined as: where is any interval of length . The logarithm of the density of the wrapped Cauchy distribution may be written as a Fourier series in : where which yields: (c.f. Gradshteyn and Ryzhik 4.224.15) and (c.f. Gradshteyn and Ryzhik 4.397.6). The characteristic function representation for the wrapped