Let \(u: \mathbb{R} ightarrow \mathbb{R}\) be a Borel measurable function and assume that \(x \mapsto e^{\lambda x} u(x)\) is integrable for each \(\lambda \in \mathbb{R}\). Show that for all \(z \in \mathbb{C}\)

\[\int_{\mathbb{R}} e^{z x} u(x) d x=\sum_{n=0}^{\infty} \frac{z^{n}}{n !} \int_{\mathbb{R}} x^{n} u(x) d x\]