We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$ We may freely use the formula $T_{0}(0) = \frac{\sqrt{\pi}}{2}$. a) Show that if $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$, the integral defining $T_{m}(x)$ is convergent. b) What is the interval $A$ of $m \in \mathbb{R}$ such that the integral defining $T_{m}(0)$ is convergent? c) Calculate $T_{2k}(0)$ and $T_{2k+1}(0)$ for $k \in \mathbb{N}$ (in terms of $k!$ and $(2k)!$).
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$
$$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
We may freely use the formula $T_{0}(0) = \frac{\sqrt{\pi}}{2}$.
a) Show that if $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$, the integral defining $T_{m}(x)$ is convergent.
b) What is the interval $A$ of $m \in \mathbb{R}$ such that the integral defining $T_{m}(0)$ is convergent?
c) Calculate $T_{2k}(0)$ and $T_{2k+1}(0)$ for $k \in \mathbb{N}$ (in terms of $k!$ and $(2k)!$).