Show that $$\forall u \in \mathbb{R}, \quad \forall a \in \mathbb{R}_+^*, \quad \mathrm{e}^{\frac{au^2}{2}} = \int_{-\infty}^{+\infty} \mathrm{e}^{ut - \frac{t^2}{2a}} \frac{\mathrm{~d}t}{\sqrt{2\pi a}}$$
Show that
$$\forall u \in \mathbb{R}, \quad \forall a \in \mathbb{R}_+^*, \quad \mathrm{e}^{\frac{au^2}{2}} = \int_{-\infty}^{+\infty} \mathrm{e}^{ut - \frac{t^2}{2a}} \frac{\mathrm{~d}t}{\sqrt{2\pi a}}$$