§35.2 Laplace Transform

Notes:: See Gårding (1947), Herz (1955, p. 479), Muirhead (1982, p. 252). See also Siegel (1935), Bochner and Martin (1948, pp. 90–92, 113–132).
Keywords:: Laplace transform, functions of matrix argument
Permalink:: http://dlmf.nist.gov/35.2

¶ Definition

Keywords:: Laplace transform

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

35.2.1 $g(\mathbf{Z})=\int _{{\boldsymbol{\Omega}}}\mathop{\mathrm{etr}\/}\nolimits\!\left(-\mathbf{Z}\mathbf{X}\right)f(\mathbf{X})d\mathbf{X},$

Symbols:: $dx$ : differential of $x$ , $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, ${\boldsymbol{\Omega}}$ : space of matrices and $f(\mathbf{X})$ : complex-valued function
Referenced by:: ¶ ‣ §35.2
Permalink:: http://dlmf.nist.gov/35.2.E1
Encodings:: TeX, pMML, png

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 $|f(\mathbf{X})|<\mathop{\mathrm{etr}\/}\nolimits\!\left(-\mathbf{X}_{0}\mathbf{X}\right)$ for all $\mathbf{X}\in{\boldsymbol{\Omega}}$ . Then (35.2.1) converges absolutely on the region $\realpart{(\mathbf{Z})}>\mathbf{X}_{0}$ , and $g(\mathbf{Z})$ is a complex analytic function of all elements $z_{{j,k}}$ of $\mathbf{Z}$ .

¶ Inversion Formula

Keywords:: Laplace transform

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

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

Symbols:: $dx$ : differential of $x$ , $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $f(\mathbf{X})$ : complex-valued function and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.2.E2
Encodings:: TeX, pMML, png

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

¶ Convolution Theorem

Keywords:: Laplace transform

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 $f_{1}*f_{2}(\mathbf{T})=\int\limits _{{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}}f_{1}(\mathbf{T}-\mathbf{X})f_{2}(\mathbf{X})d\mathbf{X}.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $f(\mathbf{X})$ : complex-valued function
Referenced by:: §35.6(ii)
Permalink:: http://dlmf.nist.gov/35.2.E3
Encodings:: TeX, pMML, png