applying Stirling’s approximation.

For $t > R$ and $N > 0$ we define $f^{+}_N (t)$ by $$f_N^{+}(t) = \frac{1}{2\pi i} \int_0 ^N e^{-t\xi} \widetilde{\mathcal{F}}_\mu (-i\xi)d\xi.$$ $f_N^{+}(t)$ converges to $f^{+}(t):= \frac{1}{2\pi i}\sum_{n=0}^\infty \frac{c_n}{t^{n+1}}$ as $N \to \infty$ locally uniformly in $(R, \infty)$.

Changing the path, we obtain $$f_N^{+}(t)  = \frac{1}{2\pi}\int_0 ^N e^{-it\xi} \mathcal{F}_\mu (\xi)d\xi -  \int_{\Gamma_N} e^{-t\xi} \widetilde{\mathcal{F}}_\mu (-i\xi)d\xi,$$ where $\Gamma_N$ is a curve defined by $\Gamma_N = \{Ne^{i\theta}; 0 \leq \theta \leq \frac{\pi}{2}\}$. The integral over the path $\Gamma_N$ converges to $0$ as $N \to \infty$ locally uniformly. Therefore, we obtain $$\frac{1}{2\pi}\int_0 ^N e^{-it\xi} \mathcal{F}_\mu (\xi)d\xi \to  \frac{1}{2\pi i}\sum_{n=0}^\infty \frac{c_n}{t^{n+1}}$$ locally uniformly in $(R, \infty)$. By summing up the complex conjugate, $$\frac{1}{2\pi}\int_{-N} ^N e^{-it\xi} \mathcal{F}_\mu (\xi)d\xi  \to  \frac{1}{\pi}\sum_{n=0}^\infty \frac{{\text{\normalfont Im~}}c_n}{t^{n+1}}$$ locally uniformly. Similarly we can prove the above convergence for $t < -R$: we have only to replace $f_N^{+}(t)$ by $f_N^{-}(t):= \frac{i}{2\pi } \int_0 ^N e^{t\xi} \widetilde{\mathcal{F}}_\mu (i\xi)d\xi$. Now Lévy’s inversion formula (see Theorem 6.2.1 in [@KLC] for instance) implies that $\mu$ has the absolutely continuous density $\frac{1}{\pi}\sum_{n=0}^\infty \frac{{\text{\normalfont Im~}}c_n}{t^{n+1}}$ for $|t| > R$, which completes the proof.

**5.** The equalities of $r_\mu(k)$ and $\widetilde{R}_\mu$ are easily proved by using the above proofs of **1**, **2**, **4** and Theorem \[thm1\] 