Confidence disc for Cauchy distributions

We will construct a confidence region of parameters for $N$ samples from Cauchy distributed random variables. Although Cauchy distribution has two parameters, a location parameter $\mu \in \mathbb{R}$ and a scale parameter $\sigma > 0$, we will infer them at once by regarding them as a single complex parameter $\gamma := \mu + i \sigma$. Therefore the region should be a domain in the complex plane and we will give a simple and concrete formula to give the region as a disc.