# if h 1 then a is nonsingular and lim xk x

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Unformatted text preview: s http://www.amazon.com/exec/obidos/ASIN/0898714540 611 Therefore, ⎛ It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] u(t) = eAt c = e−rt/V ⎞ 2 c1 + c2 (rt/V ) + c3 (rt/V ) /2 ⎝ ⎠, c2 + c3 (rt/V ) c3 and, just as common sense dictates, the pollutant is never completely ﬂushed from the tanks in ﬁnite time. Only in the limit does each ui → 0, and it’s clear that the rate at which u1 → 0 is slower than the rate at which u2 → 0, which in turn is slower than the rate at which u3 → 0. D E Example 7.9.8 The Cauchy integral formula is an elegant result from complex analysis stating that if f : C → C is analytic in and on a simple closed contour Γ ⊂ C with positive (counterclockwise) orientation, and if ξ0 is interior to Γ, then f (ξ0 ) = 1 2π i Γ f (ξ ) dξ ξ − ξ0 f (j ) (ξ0 ) = and j! 2π i T H Γ f (ξ ) dξ. (ξ − ξ0 )j +1 (7.9.21) These formulas produce analogous representations of matrix functions. Suppose that A ∈ C n×n with σ (A) = {λ1 , λ2 , . . . , λs } and index (λi ) = ki . For a complex variable ξ, the resolvent of A ∈ C n×n is deﬁned to be the matrix IG R R(ξ ) = (ξ I − A)−1 . If ξ ∈ σ (A) , then r(z ) = ( − z )−1 is deﬁned at A with r(A) = R(ξ ), so the spectral resolution theorem (p. 603) can be used to write s ki −1 R(ξ ) = i=1 j =0 Y P r(j ) (λi ) (A − λi I)j Gi = j! O C ki −1 s i=1 j =0 1 (A − λi I)j Gi . (ξ − λi )j +1 If σ (A) is in the interior of a simple closed contour Γ, and if the contour integral of a matrix is deﬁned by entrywise integration, then (7.9.21) produces 1 2π i f (ξ )(ξ I − A)−1 dξ = Γ 1 2π i 1 = 2π i s f (ξ )R(ξ )dξ Γ s Γ i=1 j =0 ki −1 = i=1 j =0 s ki −1 = i=1 j =0 Copyright c 2000 SIAM ki −1 1 2π i f (ξ ) (A − λi I)j Gi dξ (ξ − λi )j +1 Γ f (ξ ) dξ (A − λi I)j Gi (ξ − λi )j +1 f (j ) (λi ) (A − λi I)j Gi = f (A). j! Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 612 Chapter 7 Eigenvalues and Ei...
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## This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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