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By Karl W. Breitung

This e-book provides a self-contained advent to the topic of asymptotic approximation for multivariate integrals for either mathematicians and utilized scientists. a suite of result of the Laplace tools is given. Such equipment are priceless for instance in reliability, statistics, theoretical physics and knowledge conception. a big distinctive case is the approximation of multidimensional general integrals. right here the relation among the differential geometry of the boundary of the combination area and the asymptotic chance content material is derived. probably the most vital functions of those equipment is in structural reliability. Engineers operating during this box will locate right here a whole define of asymptotic approximation tools for failure chance integrals.

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Here /3 is a real parameter. We set here as parameter ~2 instead of ,~ deviating from the usual terminology, since in the applications in the following chapters for normal integrals the parameter appears in this form. In the years 1948-1966 the Laplace method was generalized to approximations for multivariate integrals. The first result is given in the article of Hsu [76] in 1948. Here the most simple case as described in theorem 41, p. 56 is treated. The most extensive monography about the Laplace method for bivariate integrals is from J.

There exist integrable functions h, hi : ~ n --+ ~ such that for r E I always If( ,r)l < h(ze) and IL-(x, T)P < hi(x). 60) 2. For all r E I the set G ( r ) = {x;g(~, r) = O} is a compact Cl-manifold in 1Rn and for all x G G(r) always ~Txg(x, r) 5s o. (An orientation for G(v) is given by the normal field nr = IVxg(x, r)[-1V~eg(x, r) ). 61) g(x,~)<0 exists for all r G I and the derivative of this integral with respect to r is given by F'(T)= j f~(x, T) dx - j g(m,~)

More detailed discussions and results can be found in [100] and [106]. L e m m a 25 /? = dx Ixl~l(~) co S . 101) PROOF: F = 2 /0 Ixl~l(x) dx = 2 d~l(x)l = 2 . ~:::~1(0) co dx / dx = 2 1--~ v~ : 9 [] L e m m a 26 Given are a negative definite n • n-matrix A and two n-dimensional vectors b and c. 103) ~/[ det(A) l I c T x [ e x p ( ~1x T A x ) d x = 30 2(27r)(n-1)/2 c T a - l c det(A) 1/2. 104) PROOF: The first two assertions are shown in [100], p. 19. To prove the last assertion, we assume that - A is the inverse of a covariance matrix of an n-dimensional normal distribution N(o, - A - l ) .

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