Download e-book for kindle: Asymptotic Approximation of Integrals by R. Wong

By R. Wong

ISBN-10: 0898714974

ISBN-13: 9780898714975

Asymptotic equipment are usually utilized in many branches of either natural and utilized arithmetic, and this vintage textual content is still the main updated e-book facing one vital point of this sector, particularly, asymptotic approximations of integrals. during this publication, all effects are proved conscientiously, and plenty of of the approximation formulation are followed by way of blunders bounds. an intensive dialogue on multidimensional integrals is given, with references supplied. Asymptotic Approximations of Integrals comprises the 'distributional method', no longer to be had somewhere else. lots of the examples during this textual content come from concrete functions. considering the fact that its booklet twelve years in the past, major advancements have happened within the basic conception of asymptotic expansions, together with smoothing of the Stokes phenomenon, uniform exponentially better asymptotic expansions, and hyperasymptotics. those new ideas belong to the realm referred to now as 'exponential asymptotics'. Expositions of those new theories are available papers released in numerous journals, yet no longer but in e-book shape.

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An alternative method is to apply the generalized Watson's lemma, and this can be done by making the substitution t = x(l — i>), followed by an integration by parts. 46) at v = \ and let It is easily shown that By writing with 28 I Fundamental Concepts of Asymptotics we also have where and F(a, x) being the complementary incomplete Gamma function defined by On making the substitution t = x(l + T), the last integral can be written as Since it follows that if a > 1 and x > a — 1. Therefore if x > 2N - 1.

68). Since f(2m\x) and f(2m +2\x) have the same sign in (0, n), and since B2m — B2m(x) is of the same sign as B2m in [0,1], the remainders Rm(ri) and Rm +a(n) obviously have opposite signs. The corollary is thus proved. Example 6. 2). 5). 18) with n replaced by n — 1. The result is where Now fix m and let n -> + oo. Since |B2m — B2m(x — [x])\ is bounded by (2 - 21_2M)|b2M|,2m|, the last integral converges if we replac This establishes the existence of the limit 6. The Euler-Maclaurin Summation Formula 39 where The number 7 is the well-known Euler constant', see Olver (1974a, p.

2) is valid. Finally, we emphasize the importance to consider, at the end of any investigation, whether or not the result obtained is worth having. 10). 14 I Fundamental Concepts of Asymptotics 4. Integration by Parts A simple yet powerful technique for deriving asymptotic expansions of definite integrals is the method of integration by parts. Each integration produces a term in the expansion, and the error term is given explicitly as an integral. This idea can be made clear by considering the following examples.

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Asymptotic Approximation of Integrals by R. Wong

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