Divergent Solutions of the Heat Equation : on an Article of Lutz , Miyake and Sch

In this article, we generalize results of Lutz, Miyake and Sch afke concerning summability of formal solutions of the Cauchy problem for the complex heat equation. In particular , we show that the type of summability depends on the given initial condition. 0. Introduction. In detail, we will be concerned with the following problem for the complex heat equation (in two complex variables and z): u (; z) = u zz (; z); u(0; z) = ^ '(z); (0.1) where for the time being ^ '(z) = P ' n z n may denote a power series which may or may not have positive radius of convergence. A straightforward (formal) computation shows that (0.1) has a unique formal power series solution (in two variables) which may be written as ^ u(; z) =