TITLE: Gaussian integral with a shift in the complex plane
QUESTION [15 upvotes]: $$ \int_{-\infty}^\infty e^{-(x+ia)^2} \text{d}x $$ 
where $a\in \mathbb{R}$.
I don't know where to start but have reasons to believe the answer is $\sqrt{\pi}$. Namely $\int_{-\infty}^\infty e^{-x^2} \text{d}x$. Problem is I feel insecure performing changes of variables when suddenly the variables range in $\mathbb{C}$. I don't even know if this can be done properly, least of all how.

REPLY [18 votes]: I saw this used once. Basically, we show that the value of the integral doesn't change with respect to $a$. It requires differentiation under the integral.
$$
I(a) = \int_{-\infty}^{\infty} \exp(-(x+ia)^2) dx
$$
\begin{align*}
\frac{dI}{da} &= \int_{-\infty}^{\infty} \frac{d}{da} \exp(-(x+ia)^2) dx \\\\
&= \int_{-\infty}^{\infty} -2i(x+ia)\exp(-(x+ia)^2) dx \\
&= i \int_{-\infty}^{\infty} \frac{d}{dx} \exp(-(x+ia)^2) dx \\
&= i \exp(-(x+ia)^2) |_{x=\pm \infty} = 0
\end{align*}