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35 Functions of Matrix ArgumentProperties

§35.2 Laplace Transform

Definition

For any complex symmetric matrix 𝐙,

35.2.1 g(𝐙)=𝛀etr(𝐙𝐗)f(𝐗)d𝐗,

where the integration variable 𝐗 ranges over the space 𝛀.

Suppose there exists a constant 𝐗0𝛀 such that |f(𝐗)|<etr(𝐗0𝐗) for all 𝐗𝛀. Then (35.2.1) converges absolutely on the region (𝐙)>𝐗0, and g(𝐙) is a complex analytic function of all elements zj,k of 𝐙.

Inversion Formula

Assume that 𝓢|g(𝐔+i𝐕)|d𝐕 converges, and also that its limit as 𝐔 is 0. Then

35.2.2 f(𝐗)=1(2πi)m(m+1)/2etr(𝐙𝐗)g(𝐙)d𝐙,

where the integral is taken over all 𝐙=𝐔+i𝐕 such that 𝐔>𝐗0 and 𝐕 ranges over 𝓢.

Convolution Theorem

If gj is the Laplace transform of fj, j=1,2, then g1g2 is the Laplace transform of the convolution f1*f2, where

35.2.3 f1*f2(𝐓)=𝟎<𝐗<𝐓f1(𝐓𝐗)f2(𝐗)d𝐗.