Read e-book online Asymptotic Approximations for Probability Integrals PDF

By Karl W. Breitung

ISBN-10: 3540586172

ISBN-13: 9783540586173

This publication supplies a self-contained advent to the topic of asymptotic approximation for multivariate integrals for either mathematicians and utilized scientists. a set of result of the Laplace equipment is given. Such equipment are valuable for instance in reliability, records, theoretical physics and data thought. an enormous precise case is the approximation of multidimensional general integrals. the following the relation among the differential geometry of the boundary of the mixing area and the asymptotic chance content material is derived. essentially the most very important purposes of those tools is in structural reliability. Engineers operating during this box will locate the following an entire define of asymptotic approximation tools for failure chance integrals.

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C o r o l l a r y 37 Let f and h be continuous functions on a finite interval [c~,fl]. 33) b) If the global maximum occurs only at ~, h(~) # 0 and f ( x ) is near c~ continuously differentiable with f ' ( c 0 < O, then 1 I(A) ~ h(o~)exp(Af(cr))Alf,(oOi , A ~ oo. 34) c) If the global maximum occurs only at ~, h(o0 # 0 and f ( x ) is twice continuously differentiable near ct with f'(c~) = 0 and f " ( ~ ) < O, then Z(,~) ~ h(c~)exp(,~f(c~)) PROOF: 2,Xlf,,(cO [ , ,~ ~ ~ . 35) T h e results follow directly from the last theorem.

37) This is shown in the next theorem. T h e o r e m 32 Let the function f have an asymptotic expansion to N terms with respect to the asypmtotic scale {r as x ---* xo in the form N f(x) ~ E a,~r x -~ x0. 38) rt~-I Then the coefficients a l , . . , am are uniquely determined. O PROOF: See [15], p. 16-17. EXAMPLE: But it is possible that different functions have the same asymptotic expansion. 40) n--~l as x --+ 0% since x '~ e x p ( - x ) tends to zero as x --+ oo for all n C ,W. [] The following two theorems about integration and differentiation are from [15].

20) T h e differentiation of order relations and a s y m p t o t i c equations is in general possible only under some restrictions. Results are given in [105], p. 8-11 for holomorphic functions and functions with m o n o t o n e derivatives. A more complete review of these results can be found in [11]. T h e integration is much easier as shown in the following lemma. 38 L e m m a 31 Is S and g are continuous functions on (a, oo) with g > O, we have for integrals the following results . IS $ ~ g(~)d~ = oo, then f [ f ( y ) d y = 0 ( ] ~ g(y)d~), ~ -~ oo, (~) f = O ( g ) , ~ - ~ oo (b) S = o(g),~ ~ 0o ~ f ( y ) d y = o(f~ g(y)dy), x --, oo, (c) f ~ c.

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Asymptotic Approximations for Probability Integrals by Karl W. Breitung

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