Characterizations of the maximum likelihood estimator of the Cauchy distribution

We consider characterizations of the maximal likelihood estimator (MLE) of samples from the Cauchy distribution. We characterize the MLE as an attractive fixed point of a holomorphic map on the upper-half plane. We show that the iteration of the holomorphic function starting at every point in the upper-half plane converges to the MLE exponentially fast. We can also characterize the MLE as a unique root in the upper-half plane of a certain univariate polynomial over $\mathbb R$. By this polynomial, we can derive the closed-form formulae for samples of size three and four, and furthermore show that for samples of size five, there is no algebraic closed-form formula.