For $a > 0$ and $b \in \mathbb{R}^d$ it is not too hard to show that $$ \int_{\mathbb{R}^d} e^{\langle b \mid x \rangle - \frac{\|x\|^2}{a}} dx = (\pi a)^{d/2}e^{\frac{a}{4}\sum\limits_{i=1}^d b_k^2}$$ Now what about $b \in \mathbb{C}^d$? I know that the formula still works out just fine in that case, but all the references I find just state that this is a simple application of analytic continuation, which is very mysterious to me to be honest. As far as I know, it can be justified by contour integration, but this seems to be quite tedious. However, there is another approach that states that the integral on the left hand side may be seen as a holomorphic function in the variable $b \in \mathbb{C}$ (if we restrict to the case were $d=1$, since knowing this case implies the case of $d>1$) and this holomorphic function agrees on some region with the holomorphic function on the right, so I guess we can somehow deduce, by using the identity theorem, that those two function are actually the same.
Can someone elaborate on how and why this is justified?