TITLE: Criteria of convergence for gaussian integral with complex parameters
QUESTION [0 upvotes]: Consider the following Gaussian integral:
\begin{equation}
I=\int_{-\infty}^{+\infty} e^{\mathrm{i}ax^2+bx}\mathrm{d}x
\end{equation}
Where $a\in \mathbb{R}$.
If $b=0$ then:
$$I=\int_{0}^{\infty} e^{i a x^{2}} d x=e^{i \pi \operatorname{sgn}(a) / 4} \sqrt{\frac{\pi}{4|a|}}$$
If $b$ is purely imaginary then (from wikipedia):
$$I=\left(\frac{ \pi i}{a}\right)^{\frac{1}{2}} e^{\frac{-i \mathrm{Im}[b]^{2}}{4a}}$$
My questions:

If $b\in \mathbb{C}$ where $\mathrm{Re}[b]\neq 0$ what are the constraints needed on $a$ so that the integral still converges?
For general $a,b\in\mathbb{C}$ How must $a$ relate to $b$ so that the integral converges? More generally, what are the general conditions for $I$ to converge? (Setting $\mathrm{Re}[a]<0$ is not the most general condition: indeed if $b=0$ and $a$ purely imaginary then $I$ still converges).

Any reference or link to a specific book/article is always appreciated. Thanks for the help.

REPLY [1 votes]: For real $c $ and $d$ $e^{cx^{2}+dx}$ is integrable iff $c <0$. Since $|e^{ia x^{2}+bx}|=e^{\Re (ia x^{2}+bx)}=e^{-\Im a x^{2}+\Re b x}$ it folows that the required condition  is $\Im a >0$.
The integral converges for all complex $b$ if the imaginary part of $a$ is positive and diverges for all complex $b$ if the imaginary part of $a$ is non-negative.
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