significant role in free probability theory is the following set $$\mathcal{F}=\left\{F|F:\mathbb{C}^+\to\mathbb{C}^+\;\;\mathrm{is}\;\;\mathrm{analytic}\;\;\mathrm{and}
\;\;\lim_{y\to\infty}\frac{F(iy)}{iy}=1\right\}.$$ A function $F$ belongs to $\mathcal{F}$ if and only if $F=F_\mu:=1/G_\mu$ for some $\mu\in\mathcal{M}$. The function $F_\mu$ is called the reciprocal Cauchy transform of $\mu$. Any function $F\in\mathcal{F}$ has the property $\Im F(z)\geq\Im z$ for $z\in\mathbb{C}^+$ and has a Nevanlinna representation of the form $$\label{NeF}
F(z)=\Re F(i)+z+\int_\mathbb{R}\frac{1+sz}{s-z}\;d\rho(s),$$ where $\rho$ is some finite positive Borel measure on $\mathbb{R}$. Moreover, the function $F$ has a right inverse $F_\mu^{-1}$ with respect to composition, which is defined on the truncated cone $$\Gamma_{\alpha,\beta}=\{x+iy\in\mathbb{C}^+:|x|\leq\alpha
y,\;|y|\geq\beta\}$$ of the upper half-plane for some $\alpha,\beta>0$. The function $\phi_\mu:\Gamma_{\alpha,\beta}\to\mathbb{C}^-\cup\mathbb{R}$ defined by $$\phi_\mu(z)=F_\mu^{-1}(z)-z,\;\;\;\;\;z\in\Gamma_{\alpha,\beta},$$ is called the Voiculescu transform of $\mu$. As indicated in the introduction, for $\mu,\nu\in\mathcal{M}$ and $z$ in some truncated cone in $\mathbb{C}^+$ the following identity holds: $$\phi_{\mu\boxplus\nu}(z)=\phi_\mu(z)+\phi_\nu(z).$$ Particularly, the identity $F_{\mu\boxplus\delta_a}(z)=F_\mu(z-a)$ holds for $z\in\mathbb{C}^+$ and $a\in\mathbb{R}$.

The reciprocal Cauchy transform $F_\mu$ can be used to locate the atoms of $\mu$. A point $\alpha$ is an atom of $\mu$ if and only if $F_\mu(\alpha)=0$ (that is, $F_\mu$ is defined and takes the value $0$ at the point $\alpha$) and the Julia-Carathéodory derivative $F_\mu'(\alpha)$ (which is the limit of $$\frac{F_\mu(z)-F_\mu(\alpha)}{z-\alpha}$$ as $z\to\alpha$ nontangentially, i.e., $(\Re
z-\alpha)/\Im z$ stays 