Let $Z$ be a real random variable, we define the Laplace transform of $Z$ as follows: $L_Z(\lambda) = \mathbb{E}(e^{\lambda Z})$ for $\lambda \in \mathbb{R}$.
What links can be shown between the Laplace transform of $Z$ and the moments of $Z$
If $Z$ has a CDF $F_Z$. The moment generating function (mgf) of $Z$ (or $F_Z$ ), denoted by $M_Z(t)$, is $M_Z(t)=\mathrm{E}(e^{t Z})$. Thus $L_Z(\lambda) = \mathbb{E}(e^{\lambda Z}) = M_Z(\lambda)$.
What links can be shown between the Laplace transform of $Z$ and the characteristic function of $Z$
We define the characteristic function of $Z$ as $\varphi_X(t)=\mathrm{E}(e^{i t X})$. Thus $L_Z(i\lambda) = \mathbb{E}(e^{i\lambda Z}) = \varphi_X(\lambda)$.
Is this correct or do I have to change something?