Analytic Solutions of the Heat Equation

Abstract

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 $t\in \mathbb{C}$.

38 pages

References in corpus (1)

• The De Bruijn-Newman constant is non-negative