By R. Wong

Asymptotic equipment are usually utilized in many branches of either natural and utilized arithmetic, and this vintage textual content continues to be the main updated publication facing one very important element of this region, specifically, asymptotic approximations of integrals. during this booklet, all effects are proved carefully, and lots of of the approximation formulation are followed via mistakes bounds. a radical dialogue on multidimensional integrals is given, with references supplied. Asymptotic Approximations of Integrals comprises the 'distributional method', now not on hand in different places. many of the examples during this textual content come from concrete functions. considering its booklet twelve years in the past, major advancements have happened within the common idea of asymptotic expansions, together with smoothing of the Stokes phenomenon, uniform exponentially more desirable asymptotic expansions, and hyperasymptotics. those new ideas belong to the realm referred to now as 'exponential asymptotics'. Expositions of those new theories come in papers released in quite a few journals, yet no longer but in ebook shape.

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**Sample text**

16), we know that the sign of B2m — B2m(%) is the same as the sign of B2m. By Lemma 1 (ii) and (iii), we conclude that the same is true for B2m — B2m(x) in [0,1]. 20). 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.

32). 34). 35) to theh function f(x) = l/(* + 2). 32). 14) and the error test, we have Example 8. 6), we first note the Fourier cosine series expansion 6. The Euler-Maclaurin Summation Formula 41 see Tolstov (1962, p. 26). 43) can be evaluated by using the identity which can be obtained by summing the series eix + ei3x + ... + ei(2k-1)x and then taking the imaginary part. 44), it follows that The above ingenious argument is due to Szego (1921). 45), and proceeds as follows. 45). 9). 4). 48). 49).

71) is not valid. 71). Converse to Watson's Lemma. 68). // as 2 -* oo uniformly in |arg(z — c)|