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 .
Assume that converges, and also that . Then
where the integral is taken over all such that and ranges over .
If is the Laplace transform of , , then is the Laplace transform of the convolution , where