[math] \newcommand{\mathds}{\mathbb}[/math]

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Draw the picture of the following function, and of its inverse,

[[math]] f(z)=\frac{z+ir}{z-ir} [[/math]]

with [math]r\in\mathbb R[/math], and prove that for [math]r \gt \gt 0[/math] and [math]T=T^*[/math], the element [math]f(T)[/math] is well-defined.