Solutions of Exponential Growth to Systems of Partial Differential Equations

This result is very deep and its proof rather long and intricate. In the present paper we show that if one restricts attention to C ~ functions on R n which are of exponential growth, it is possible to derive the formula by classical methods. Moreover, it is possible to obtain the measures featuring in the representation explicitly in terms of the represented solution. A's a fringe benefit we also obtain what "Goursat data" on the boundary of the ordant [0, ~)~ (or any other ordant) is sufficient to determine the values of the solution in the interior of that ordant. Our tools are all old-fashioned: Laplac~ transform, Green's identity and residues. The discrete analog of the present paper, i.e. the case where partial differential equations are replaced by partial difference equations, has been done in Zeilberger [4], which is a prerequisite. In fact, since most of the proofs are similar, we shall not give complete proofs, but instead describe how t o adapt the proofs of [4] to the present situation. We are going to prove the following