R. Wong's Asymptotic and computational analysis: conference in honor PDF

By R. Wong

ISBN-10: 0585332770

ISBN-13: 9780585332772

ISBN-10: 0824783476

ISBN-13: 9780824783471

Papers offered on the foreign Symposium on Asymptotic and Computational research, held June 1989, Winnipeg, Man., subsidized by means of the department. of utilized arithmetic, college of Manitoba and the Canadian utilized arithmetic Society.

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We can easily establish that f (t) = (1/a)[1 − e−at ]. 42 CHAPTER 2. INVERSION FORMULAE AND PRACTICAL RESULTS Thus lim f (t) = 1/a. t→∞ Application of the theorem gives lim sf¯(s) = lim 1/(s + a) = 1/a, s→0 s→0 which confirms the result obtained previously. Also, applying the initial value theorem gives lim sf¯(s) = lim 1/(s + a) = 0 = f (0), s→∞ s→∞ which confirms the result that lim f (t) = 0. 5 Series and Asymptotic Expansions In the previous section we were interested in estimating the limiting value of f (t), if it existed, as t → 0 and t → ∞.

Thus the function p(x) = {1 + (1/c)(b − x)(x − a)}r , can be made as large as we please in a < x < b and as small as we like in 0 < x < a, b < x < 1 by appropriate choice of r. But p(x) is a polynomial in x, and by our hypothesis 1 xn−1 ψ(x)dx = 0, for n = 1, 2, · · · 0 we should have 1 p(x)ψ(x)dx = 0, 0 for every positive integer r. But the above inequalities imply that, by choosing r large enough, 1 p(x)ψ(x)dx > 0. 0 The first alternative thus leads to a contradiction. A similar argument applies if we assume ψ(x) < 0 in (a,b).

8 Multidimensional Laplace Transforms We have restricted our exposition of Laplace transforms to functions of one variable but we could equally well extend our definition to a function of two or more variables. e. 1, — see Ditkin and Prudnikov [67] for a fuller account of the criteria needed for convergence. We can, as earlier, deduce the Laplace transform for elementary functions of two variables. Useful general results are:1. If f (t1 , t2 ) = f1 (t1 )f2 (t2 ) and L{f1 (t1 )} = f¯1 (s1 ) and L{f2 (t2 )} = f¯2 (s2 ) then L2 {f (t1 , t2 )} = f¯1 (s1 )f¯2 (s2 ).

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Asymptotic and computational analysis: conference in honor of Frank W.J. Olver's 65th birthday by R. Wong


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