35 Functions of Matrix Argument Properties 35.1 Special Notation 35.3 Multivariate Gamma and Beta Functions

§35.2 Laplace Transform

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Keywords:: Laplace transform, for functions of matrix argument, functions of matrix argument
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).
Referenced by:: Erratum (V1.0.27) for Paragraph Inversion Formula (in §35.2)
Permalink:: http://dlmf.nist.gov/35.2
See also:: Annotations for Ch.35

Definition

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Keywords:: Laplace transform, analytic properties, definition, for functions of matrix argument
See also:: Annotations for §35.2 and Ch.35

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

35.2.1		$g(\mathbf{Z})=\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{Z}% \mathbf{X}\right)f(\mathbf{X})\,\mathrm{d}{\mathbf{X}},$
ⓘ Defines: $g$ : Laplace transformation of $f$ (locally) Symbols: $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $\mathbf{Z}$ : complex symmetric $m\times m$ matrix and $f(\mathbf{X})$ : complex-valued function of matrix argument Keywords: Laplace transform Referenced by: §35.2 Permalink: http://dlmf.nist.gov/35.2.E1 Encodings: TeX, pMML, png See also: Annotations for §35.2, §35.2 and Ch.35

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

Inversion Formula

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Keywords:: Laplace transform, for functions of matrix argument, inversion formula
Referenced by:: Erratum (V1.0.27) for Paragraph Inversion Formula (in §35.2)
Clarification (effective with 1.0.27):: The wording of the first sentence was changed to make the integration variable more apparent.
See also:: Annotations for §35.2 and Ch.35

Assume that $\int_{\boldsymbol{\mathcal{S}}}\left|g(\mathbf{U}+\mathrm{i}\mathbf{V})\right|% \,\mathrm{d}{\mathbf{V}}$ converges, and also that its limit as $\mathbf{U}\to\infty$ is $0$ . Then

35.2.2		$f(\mathbf{X})=\dfrac{1}{(2\pi\mathrm{i})^{m(m+1)/2}}\int\operatorname{etr}% \left(\mathbf{Z}\mathbf{X}\right)g(\mathbf{Z})\,\mathrm{d}{\mathbf{Z}},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\,\mathrm{d}$ : differential of matrix, $\mathbf{Z}$ : complex symmetric $m\times m$ matrix, $f(\mathbf{X})$ : complex-valued function of matrix argument, $m$ : positive integer and $g$ : Laplace transformation of $f$ Permalink: http://dlmf.nist.gov/35.2.E2 Encodings: TeX, pMML, png See also: Annotations for §35.2, §35.2 and Ch.35

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

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Keywords:: Laplace transform, convolution theorem, for functions of matrix argument
See also:: Annotations for §35.2 and Ch.35

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		$f_{1}*f_{2}(\mathbf{T})=\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}f_% {1}(\mathbf{T}-\mathbf{X})f_{2}(\mathbf{X})\,\mathrm{d}{\mathbf{X}}.$
ⓘ Defines: $*$ : convolution operator (locally) Symbols: $\int$ : integral, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\,\mathrm{d}$ : differential of matrix, $<$ : less-than for matrices, $f(\mathbf{X})$ : complex-valued function of matrix argument and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros Referenced by: §35.6(ii) Permalink: http://dlmf.nist.gov/35.2.E3 Encodings: TeX, pMML, png See also: Annotations for §35.2, §35.2 and Ch.35

35.1 Special Notation 35.3 Multivariate Gamma and Beta Functions

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§35.2 Laplace Transform

Definition

Inversion Formula

Convolution Theorem

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