Although this is more of a mathematical question, I will in what follows refer to an answer of @Qmechanic that has been posted in this forum (I am sorry for creating a new post for this, but I actually have two different questions and I also did not earn enough points to comment on posts).
I am trying to prove two things:
- the divergence of
$$\int_\mathbb{R}{\mathrm{d}x \, \mathrm{e}^{-a x^2}}, \qquad a \in \mathbb{C}, \quad \mathrm{Re}(a)<0;$$
- the convergence of (without a regulator)
$$\int_\mathbb{R}{\mathrm{d}x \, \mathrm{e}^{-a x^2}} = \lim_{\substack{x_i \to -\infty \\ x_f \to \infty}}\int_{x_i}^{x_f}{\mathrm{d}x \, \mathrm{e}^{-a x^2}}, \qquad a \in \mathbb{C}, \quad \mathrm{Re}(a)=0, \quad \mathrm{Im}(a) \neq 0,$$ where the first integral in "2." is to be understood in the improper sense, as stated afterwards -- see also the aforementioned post.
Regarding my first question, I claim to know a proof for $a \in \mathbb{R}$ -- as has (probably) been shown already on StackExchange, see e.g. here for the second part -- namely ($a>0$)
$$ \int_{-\infty}^{\infty}{\mathrm{d}x \, \mathrm{e}^{ax^2}} > \int_{-\infty}^{\infty}{\mathrm{d}x \, 1} \to \infty, $$
while
$$ \int_{-\infty}^{\infty}{\mathrm{d}x \, \mathrm{e}^{-ax^2}} = 2 \int_{0}^{\infty}{\mathrm{d}x \, \mathrm{e}^{-ax^2}} = 2 \left[ \int_0^1{\mathrm{d}x \, \mathrm{e}^{-ax^2}} + \int_1^{\infty}{\mathrm{d}x \, \mathrm{e}^{-ax^2}} \right] < 2 \left[ \int_0^1{\mathrm{d}x \, \mathrm{e}^{-ax^2}} + \int_1^{\infty}{\mathrm{d}x \, \mathrm{e}^{-ax}} \right] = 2 \left[ \int_0^1{\mathrm{d}x \, \mathrm{e}^{-ax^2}} - \frac{1}{a}\mathrm{e}^{-ax} \bigg\rvert_1^{\infty} \right] < \infty.$$
Furthermore, it is easy to see that for $a \in \mathbb{C}$ with $\mathrm{Re}(a)>0$, one has
$$ \left \lvert \int_{\mathbb{R}}{\mathrm{d}x \, \mathrm{e}^{-ax^2}} \right \rvert \leq \int_{\mathbb{R}}{\mathrm{d}x \, \left \lvert \mathrm{e}^{-ax^2} \right \rvert} = \int_{\mathbb{R}}{\mathrm{d}x \, \mathrm{e}^{-\mathrm{Re}(a)x^2}},$$
which converges by what is written above. However, I have no idea how to find something like a lower bound for $a \in \mathbb{C}$ with $\mathrm{Re}(a) <0$ (although it appears certainly clear to me that the integral has to diverge because the additional complex part is simply oscillating and of magnitude unity).
Regarding my second question, I actually have no idea how to start, since I did not catch the argument that was given by @Qmechanic in said post.
I appreciate any help.
