In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$. We assume that $E$ is not empty and that $f$ admits near 0 the following limited expansion of order $n \in \mathbb{N}$: $$f(t) = \sum_{k=0}^{n} \frac{a_k}{k!}t^k + O\left(t^{n+1}\right).$$ IV.C.1) Show that for all $\beta > 0$, we have, as $x$ tends to $+\infty$, the following asymptotic expansion: $$\int_0^{\beta} \left(f(t) - \sum_{k=0}^{n} \frac{a_k}{k!}t^k\right)e^{-tx}\,dt = O\left(x^{-n-2}\right).$$ IV.C.2) Deduce from this that as $x$ tends to infinity, we have the asymptotic expansion: $$Lf(x) = \sum_{k=0}^{n} \frac{a_k}{x^{k+1}} + O\left(x^{-n-2}\right).$$
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$. We assume that $E$ is not empty and that $f$ admits near 0 the following limited expansion of order $n \in \mathbb{N}$:
$$f(t) = \sum_{k=0}^{n} \frac{a_k}{k!}t^k + O\left(t^{n+1}\right).$$
IV.C.1) Show that for all $\beta > 0$, we have, as $x$ tends to $+\infty$, the following asymptotic expansion:
$$\int_0^{\beta} \left(f(t) - \sum_{k=0}^{n} \frac{a_k}{k!}t^k\right)e^{-tx}\,dt = O\left(x^{-n-2}\right).$$
IV.C.2) Deduce from this that as $x$ tends to infinity, we have the asymptotic expansion:
$$Lf(x) = \sum_{k=0}^{n} \frac{a_k}{x^{k+1}} + O\left(x^{-n-2}\right).$$