Showing that $\int_{0}^{\infty}k(x,z)f(x)dx$ is analytic with $k$ heat kernel.

Question

Let $k(x,z)=\frac{1}{(4\pi z)^{n/2}}e^{|x|/{4z}}$ heat kernel. Show that $F(z):=\int_{0}^{\infty}k(x,z)f(x)dx$ is analytic, where $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R}.)$

Will there be some theorem that allows us to conclude that, when $k (x, z)$ is analytical, then $F(z)$ is also analytical?