Analytic Solutions of the Heat Equation

Motivated by the recent proof of Newman's conjecture \cite{R-T} we study certain properties of entire caloric functions, namely solutions of the heat equation $\partial_t F = \partial_z^2 F$ which are entire in $z$ and $t$. As a prerequisite, we establish some general properties of the order and type of an entire function. Then, we start our inquiry on entire caloric functions by determining the necessary and sufficient condition for a function $f(z)$ to be the initial condition of an entire solutions of the heat equation and, subsequently, we examine the relation of the $z$-order and $z$-type of an entire caloric function $F(t, z)$, viewed as function of $z$, to its $t$-order and $t$-type respectively, if it is viewed as function of $t$. After that, we shift our attention to the zeros $z_k(t)$ of an entire caloric function $F(t, z)$, viewed as function of $z$. We show that the points $(t, z)$ at which $F(t, z) = \partial_z F(t, z) = 0$ form a discrete set in $\mathbb{C}^2$ and we derive the $t$-evolution equations of the zeros of $F(t, z)$. These are differential equations which hold for all but countably many $t \in \mathbb{C}$.