Theorems On nth Dimensional Laplace Transform

Abstract

Let U be the set of all functions from [0,∞)n to R and V be the set of all functions from S ⊆Cn to C.Then the nth dimensional Laplace transform is the mapping
Ln : U→V
defined by:
Z˜
f(˜sn) = L {F(x˜n),˜ F(x˜n)e−(˜sn·x˜n)dx˜n n
R
Where F(x˜n) ∈U and ˜sn ∈Cn
In this paper we gave alternative proof for some theorems on properties of nth dimensional Laplace Transform, we proved that if F(x˜n) is piecewise continuous on [0,∞)n and function of exponential order ˜γn = (γ1,γ2,...,γn) then the nth dimensional Laplace Transform defined above exists, absolutely and uniformly convergent, analytic and infinitely differentiable on Re(s1) > γ1,Re(s2) > γ2,...,Re(s) > γn, and we gave also some corollaries of these results.