and the Stieltjes transform.

(Complex moments). We define the $n$th *complex moment* of $\mu \in \mathcal{P}_1$ by $m_n(\mu) = m_n(\nu) + m_n(\lambda)$, where $\nu$ and $\lambda$ are given by a decomposition in (\[decomp\]). Similarly we define the first and second moments of $\mu \in \mathcal{P}_2$ by $m_j(\mu) = m_j(\nu) + m_j(\lambda)$ for $j = 1$, $2$.

\(1) The decomposition in (\[decomp\]) and the choice of the convergence radius $R$ of $\lambda$ are not unique. It is however not difficult to prove that the definition of complex moments does not depend on these ambiguities.\
(2) We notice that the domain $(0, \infty)$ of the Fourier transform and the domain ${\mathbb{C}_{-}}$ of the Stieltjes transform play analogous roles in the analytic continuations. The information of $G_\mu$ on $\operatorname*{\mathbb{C}_{+}}$ is recovered from the relation $G_\mu(z) = \overline{G_\mu(\bar{z})}$; this is similar to the relation $\mathcal{F}_\mu(z) = \overline{\mathcal{F}_\mu(-z)}$.

\(1) $\mu(dx) = \frac{2}{\pi(1+4x^4)}dx$ on $\operatorname*{\mathbb{R}}$. Then $\mathcal{F}_\mu(z) = \sqrt{2}{\text{\normalfont Re~}}\! (e^{\frac{i-1}{2}z}e^{-\frac{\pi i}{4}}) = e^{-\frac{z}{2}}(\cos(\frac{z}{2}) + \sin(\frac{z}{2}))$ for $z > 0$. The $n$th complex moment is $m_n(\mu)=i^n 2^{\frac{1-n}{2}}\sin(\frac{1-n}{4}\pi)$. In particular, $m_1(\mu)=0$, $m_2(\mu) = \frac{1}{2}$, $m_3(\mu)=\frac{i}{2}$.\
(2) $\mu_0(dx) = \frac{\sqrt{2}x^2}{\pi(1+x^4)}dx$ on $\operatorname*{\mathbb{R}}$. Then $\mathcal{F}_{\mu_0}(z) = \sqrt{2}{\text{\normalfont Re~}}\! (e^{\frac{\pi i}{4}}\exp(ize^{\frac{\pi i }{4}})) = \sqrt{2} \sum_{n=0}^\infty \frac{\cos(\frac{3n +1}{4}\pi)}{n!}z^n$ for $z 