Asymptotic Approximation of Integrals (Classics in Applied by R. Wong

By R. Wong

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The result in (ii) follows from the so-called error test, which states that if consecutive error terms associated with a series expansion have opposite signs, then each error term is numerically less than, and has the same sign as, the first neglected term of the series. This result is easy to prove, and is given in Steffensen (1950, p. 4) and Olver (1974a, p. 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.

However, to obtain results that are absolutely reliable for numerical computations, it is necessary to construct a sharp upper bound for the remainder term. Thus, if it is possible to obtain such an upper bound, then we must not neglect to do so. Example 2. Fourier Integrals. A second type of integral to which the method of integration by parts can be applied is the Fourier integral where (a, 6) is a real finite interval and f(t) is an AT-times continuously differentiable function in [a, 6]. 9) is the asymptotic expansion of F(x) as far as terms of order x ~N.

7) are called compound asymptotic expansions. It is easy to give examples of functions that have generalized asymptotic expansions but are not of Poincare type. Let Q be the set of 12 I Fundamental Concepts of Asymptotics real numbers x > c > 1. 8) is certainly an asymptotic expansion with respect to the asymptotic sequence {x~n}. However, it is not of Poincare type since the sequence q>n(x) = sin(n + l)nx/xn does not satisfy (pn +l = o((pn), as x -»• oo. For instance, q>l = sin 2nxjx — 0, when x = |m, where m is any nonnegative integer, but 2 ^ o( oo.

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