We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$ a) Let $m \in A$. Show that $T_{m}$ is continuous on $\mathbb{R}_{+}$. b) Let $m \in \mathbb{R}$. Show that $T_{m}$ is continuous on $\mathbb{R}_{+}^{*}$. c) Show that for $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$, $$T_{m}(x) \geq e^{-1} \int_{0}^{1} t^{m} e^{-x/t} dt$$ Deduce the value of $\lim_{x \rightarrow 0^{+}} T_{m}(x)$ when $m \notin A$, using the change of variable $w = x/t$.
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$
$$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $m \in A$. Show that $T_{m}$ is continuous on $\mathbb{R}_{+}$.
b) Let $m \in \mathbb{R}$. Show that $T_{m}$ is continuous on $\mathbb{R}_{+}^{*}$.
c) Show that for $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$,
$$T_{m}(x) \geq e^{-1} \int_{0}^{1} t^{m} e^{-x/t} dt$$
Deduce the value of $\lim_{x \rightarrow 0^{+}} T_{m}(x)$ when $m \notin A$, using the change of variable $w = x/t$.