  There recently has been some interest in the space of functions on an
interval satisfying the heat equation for positive time in the interior of this
interval. Such functions were characterised as being analytic on a square with
the original interval as its diagonal. In this short note we provide a direct
argument that the analogue of this result holds in any dimension. For the heat
equation on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ at
positive time all solutions are analytically extendable to a geometrically
determined subdomain $\mathcal{E}(\Omega)$ of $\mathbb{C}^d$ containing
$\Omega$. This domain is sharp in the sense that there is no larger domain for
which this is true. If $\Omega$ is a ball we prove an almost converse of this
theorem. Any function that is analytic in an open neighborhood of
$\mathcal{E}(\Omega)$ is reachable in the sense that it can be obtained from a
solution of the heat equation at positive time. This is based on an analysis of
the convergence of heat equation solutions in the complex domain using the
boundary layer potential method for the heat equation. The converse theorem is
obtained using a Wick rotation into the complex domain that is justified by our
results. This gives a simple explanation for the shapes appearing in the
one-dimensional analysis of the problem in the literature. It also provides a
new short and conceptual proof in that case.
