natural numbers.

\(c) and (d) are generalizations of (a) and (b) respectively. In (a), we can define $\nu$ to be the restriction of the Cauchy distribution $\mu_{a,b}$ on $[-R, R]$, and then $\mu_{a,b} = \nu + \lambda$ belongs to $\mathcal{P}_1$.

We check the property (A2) for the Cauchy distribution $\mu_{0, 1}$ for instance. The density function is given by $$\begin{split}
\frac{1}{\pi(1+x^2)} &= \frac{1}{\pi x^2(1 + \frac{1}{x^2})} \\
                     &= \frac{1}{\pi}\sum_{n = 0} ^\infty \frac{(-1)^n}{x^{2n+2}}.   
\end{split}$$ This series converges for $|x| > 1$.

The free, Boolean and monotone convolutions are all characterized by using Stieltjes transforms. Therefore, we consider the analytic continuation of Stieltjes transforms.

\[prop1\] Let $\lambda$ be a positive measure on $(-\infty,-R] \cup [R, \infty)$ which satisfies (A2). Then $G_\lambda (z)$ defined in ${\mathbb{C}_{-}}$ has analytic continuation to $\operatorname*{\mathbb{C}}_{-} \cup \{z \in \operatorname*{\mathbb{C}}; |z| > R \}$. We denote it by $\widetilde{G}_\lambda$. $\widetilde{G}_\lambda(z)$ has convergent series of the form $$\widetilde{G}_\lambda(z) = \sum_{n=0}^\infty \frac{b_n}{z^{n+1}}, ~~~|z| > R.$$

By assumption $\frac{1}{w^2}p(\frac{1}{w})$ is analytic in $\{z; |z| < r^{-1} \}$. We define $v = \frac{1}{z}$ for ${\text{\normalfont Im~}}z < 0$. We notice that ${\text{\normalfont Im~}}v > 0$. Then it holds that $$\notag
\begin{split}
G_\lambda(z) &= \int_{|x|>R} \frac{p(x)}{z-x}dx \\ 
           & = \int_{-R^{-1}}^{R^{-1}} \frac{p(\frac{1}{u})}{z-\frac{1}{u}}\frac{1}{u^2}du \\ 
           