In this article,we consider the integral representation of harmonic functions.Using a property of the modified Poisson kernel in a half plane,we prove that a harmonic function u(z) in a half plane with its positive part u+(z)=max{u(z),0} satisfying a slowly growing condition can be represented by its integral of a measure on the boundary of the half plan.
A necessary and sufficient condition is obtained for the incompleteness of complex exponential system in the weighted Banach space Lαp = {f：∫＋∞∞ ｜f（t）e-α（t）｜pdt ＋∞},where 1 ≤ p ＋∞ and α（t） is a weight on R.
A sufficient condition is obtained for the minimality of the complex exponential system E（A, M） = {z^le^λnz： l = 0, 1,,.., mn - 1; n = 1, 2,...} in the Banaeh space La^p consisting of all functions f such that f^-a ∈ LP（N）. Moreover, if the incompleteness holds, each function in the closure of the linear span of exponential system E（A, M） can be extended to an analytic function represented by a Taylor-Dirichlet series.
