§35.2 Laplace Transform

Definition

For any complex symmetric matrix $\mathbf{Z}$,

 35.2.1 $g(\mathbf{Z})=\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{Z}\mathbf{X% }\right)f(\mathbf{X})\mathrm{d}\mathbf{X},$

where the integration variable $\mathbf{X}$ ranges over the space ${\boldsymbol{\Omega}}$.

Suppose there exists a constant $\mathbf{X}_{0}\in{\boldsymbol{\Omega}}$ such that $|f(\mathbf{X})|<\mathrm{etr}\left(-\mathbf{X}_{0}\mathbf{X}\right)$ for all $\mathbf{X}\in{\boldsymbol{\Omega}}$. Then (35.2.1) converges absolutely on the region $\Re(\mathbf{Z})>\mathbf{X}_{0}$, and $g(\mathbf{Z})$ is a complex analytic function of all elements $z_{j,k}$ of $\mathbf{Z}$.

Inversion Formula

Assume that $\int_{\boldsymbol{\mathcal{S}}}\left|g(\mathbf{Z})\right|\mathrm{d}\mathbf{V}$ converges, and also that $\lim_{\mathbf{U}\to\infty}\int_{\boldsymbol{\mathcal{S}}}\left|g(\mathbf{Z})% \right|\mathrm{d}\mathbf{V}=0$. Then

 35.2.2 $f(\mathbf{X})=\dfrac{1}{(2\pi\mathrm{i})^{m(m+1)/2}}\int\mathrm{etr}\left(% \mathbf{Z}\mathbf{X}\right)g(\mathbf{Z})\mathrm{d}\mathbf{Z},$

where the integral is taken over all $\mathbf{Z}=\mathbf{U}+\mathrm{i}\mathbf{V}$ such that $\mathbf{U}>\mathbf{X}_{0}$ and $\mathbf{V}$ ranges over $\boldsymbol{\mathcal{S}}$.

Convolution Theorem

If $g_{j}$ is the Laplace transform of $f_{j}$, $j=1,2$, then $g_{1}g_{2}$ is the Laplace transform of the convolution $f_{1}*f_{2}$, where

 35.2.3 $f_{1}*f_{2}(\mathbf{T})=\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}f_% {1}(\mathbf{T}-\mathbf{X})f_{2}(\mathbf{X})\mathrm{d}\mathbf{X}.$