THEOREMS ON ASSOCIATION OF VARIABLES IN MULTIDIMENSIONAL LAPLACE TRANSFORMS

The inverse of the multidimensional Laplace transform is often obtained by the method of association of variables. In this paper, some basic theorems are developed for evaluating the associated transform of certain types of transformed functions. Many useful associated pairs can be produced with the aid of these fundamental theorems. Several illustrative examples are included.

In certain types of systems analysis, particularly in Volterra series applications [1][2] on non-linear systems [3][4][5], it becomes essential to invert the n-dimensional Laplace transform and specify the inverse image at a single variable, t.W denote this image function of one variable as g(t) f(tl,t 2 tn) Itl=tZ=...=tr,=t (1.2) One appreach to obtain the time function, g(t), is to associate with F(Sl,S Sn , function G(s) from which an application of the one-dimensional inverse transform yields g(t).This particular approach is called the Association of Variables.The function G(s) is said to be the associated transform of F(Sl,S 2 Sn).
Chert and Chiu [6] and Koh [7] have presented several theorems for evaluating G(s) for certain types of F(Sl,S 2 Sn).In this paper, some additional theorems are developed.Few examples are also included for ech theorem.However, once the fundamental theorems are established, it is possible to derive as many associated pairs as one desires, and use them flexibly.

2.
THEOREMS ON ASSOCIATION OF VARIABLES Suppose G(s) be the associated transform of F(Sl,S 2 s n) and Gl(S) be that of F(Sl,S 2 Sm_l,Sm+ 1 Sn), m <-n.Let k be any constant, and we restrict the variables s, s 1, s 2, s n to the right half of the complex plane.
Theorem 2.1.If a given function F(Sl,S 2 s n) can be written in the form k F(Sl,S 2 s n) Sm{Sm+a) F I s I ,s 2 Sm_ I ,Sm+ [Gl(S)-G1(s+a)] where A n means the association process for finding G(s) from F(Sl,S 2 Sn).
has the similar meaning.

CONCLUSIONS.
Theorems on associated transform developed in this paper are rigorous and very useful in performing the inverse Laplace transform for certain functions.These theorems can be applied to directly derive many associated pairs, and thus one can easily extend the tables given in [5]-[7] many fold.Moreover, the results of this paper will help develop more basic theorems in this direction, and will appear in subsequent papers.