&= \lim_{\varepsilon \searrow 0} \int_{C_{R}} e^{i\frac{z}{u}}
p\Big(\frac{1}{u}\Big)\frac{1}{u^2}du\\ 
&= \int_{C_{R}}e^{i\frac{z}{u}}
p\Big(\frac{1}{u}\Big)\frac{1}{u^2}du, 
\end{split}$$ where $\Gamma_{R, \varepsilon}$ is a curve consisting of two line segments $[-R^{-1}, -\varepsilon]$ and $[\varepsilon, R^{-1}]$ and a semicircle $C_{\varepsilon^{-1}}$. Therefore we have $$\mathcal{F}_\lambda (z) = \int_{C_{R}}e^{i\frac{z}{u}}
p\Big(\frac{1}{u}\Big)\frac{1}{u^2}du,$$ for $z > 0$. This expression continues $\mathcal{F}_\lambda$ from $(0, \infty)$ to $\operatorname*{\mathbb{C}}$.

It is natural to define what we call complex moments of $\lambda$ by the sequence $b_n$ for free, Boolean and monotone probability theories and by $c_n$ for tensor probability theory. There is no a priori evidence that these two ways define the same quantities. However, the two analytic continuations give the same complex moments as is understood from the proofs:

\[thm1\] Let $b_n$ and $c_n$ be complex numbers defined in the propositions \[prop1\] and \[prop2\]. Then $b_n = c_n$ for all $n \geq 0$ and they are expressed as $$b_n = c_n = \int_{\gamma_R} z^n p(z)dz.$$ where $\gamma_R =\{Re^{i \theta}; 0 \leq \theta \leq \pi \}$ equipped with counterclockwise direction. We denote $b_n$ by $m_n(\lambda)$.

We have proved that $b_n = c_n =\int_{C_R}\frac{p(\frac{1}{w})}{w^{n+2}}dw$. We can easily prove that this integral is equal to $\int_{\gamma_R} z^n p(z)dz$.

Therefore we can generalize moments both in terms of the Fourier transform 