${\mathbb{C}_{-}}$ to $\operatorname*{\mathbb{C}}\backslash \{a + ib \}$. We denote this function by $\widetilde{G}_{\mu_{a,b}}$. It is important that $G_{\mu_{a,b}} \neq \widetilde{G}_{\mu_{a,b}}$ in $\operatorname*{\mathbb{C}_{+}}$. We then have the convergent series $$\begin{split}
\widetilde{G}_{\mu_{a,b}}(z) &= \frac{1}{z(1 - \frac{a+ib}{z})} \\
          &= \sum_{n=0} \frac{(a+ib)^n}{z^{n+1}} 
          \end{split}$$ for $|z| > \sqrt{a^2 + b^2}$. At first sight a similar idea seems impossible for the Fourier transform because of the absolute value of $z$. However, if we restrict the domain to $z > 0$, then $\mathcal{F}_{\mu_{a,b}} (z)= e^{iaz - bz} = e^{i(a+ib)z}, ~~z > 0$. We denote its analytic continuation to $\operatorname*{\mathbb{C}}$ by $\widetilde{F}_{\mu_{a,b}}$: $$\widetilde{F}_{\mu_{a,b}} = \sum_{n=0} \frac{(a+ib)^n}{n!}(iz)^n.$$ Since the same coefficient $(a+ib)^n$ appears in both Fourier and Stieltjes transforms, we decide to call $m_n(\mu_{a,b}):=(a+bi)^n$ the *complex moments* of the Cauchy distribution. In the next section, we extend the concept of complex moments to more general probability measures.

This paper is organized as follows. The most important class of probability measures to be considered is $\mathcal{P}_1$ which roughly consists of probability measures with analytic densities at infinity. In Section \[Genmom\], we clarify similarity between the Fourier transform and the Stieltjes transform regarding analytic properties, extending the definition of moments and four kinds of cumulants to $\mathcal{P}_1$. More precisely, we generalize 