  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.
