# §35.2 Laplace Transform

## ¶ Definition

For any complex symmetric matrix ,

where the integration variable ranges over the space .

Suppose there exists a constant such that for all . Then (35.2.1) converges absolutely on the region , and is a complex analytic function of all elements of .

## ¶ Inversion Formula

Assume that converges, and also that . Then

where the integral is taken over all such that and ranges over .

## ¶ Convolution Theorem

If is the Laplace transform of , , then is the Laplace transform of the convolution , where

35.2.3