I would like to know whether the following statement is true.
Conjecture: For all $\mathbf{y} \in \mathbb{R}^d$, $s_1>0,\ldots,s_d>0$, $f_{\mathbf{y}}(\mathbf{x}) := \exp \left(-\sum_{i=1}^d \frac{(x_i - y_i)^2}{s_i^2} \right) $ is analytic in $\mathbf{x}$ on $\mathbb{R}^d$.
Apology if this is a very well known fact. I am not working in this area.
If it is true, could you please give me a reference? If this is not known, could you please let me know how I might show it? So far, I have been trying to show it by using Proposition 2.2.10 in A Primer of Real Analytic Functions which gives a condition on the partial derivatives for a function on $\mathbb{R}^d$ to be analytic. I have not yet succeeded.
Another thing I notice is that if $d=1$, then $f_y(x) = \exp\left( -(x-y)^2/s^2\right)$ is known as a Gaussian function. A Gaussian function is known to be analytic. In our case, we have a product $d$ Gaussian functions. That is, $f_\mathbf{y}(\mathbf{x})$ is separately analytic on each coordinate of $\mathbf{x}$. I would like to know if it is jointly analytic in $\mathbf{x}$.
Thanks. (This is not a homework question.)