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16), This completes the proof of the theorem. 38 I Fundamental Concepts of Asymptotics Corollary 1. 18), and has the same sign. 18). Proof. 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.

B3(X) ALLargument can now be repeated, and shows thath have the same property. This is impossible since has only one zero inside (0,1), which occurs at x = |. Therefore, the only zeros of b2S+1(X) IN [0,1]B are at x = 0, \, and 1. 7), and the second statement in (ii) is proved by an argument entirely similar to the one given above. To prove (iii), we only need observe that the extreme value of B2S(X)- b2S can occur only at its critical point x = |. This completes the proof of the lemma. )5a(x) in the interval [0,1).

As 2 -* oo uniformly in |arg(z — c)| oo, then as t -> 0+ Proof. 69) holds for any a > c. 72) that there exists a constant KN such that | for all z in Re z > c. The inversion formula gives For any fixed t in (0,1), we may choose a = c/t and put z = (c/t)(l + it). If AN= 1, then Thus The above argument is given in E. J. Watson (1981, p. 89). 32 I Fundamental Concepts of Asymptotics 6. The Euler-Maclaurin Summation Formula In the study of analytic number theory, one often encounters approximation problems concerning sums of the form ^ «;„/(&)• The best known tool for this type of problem is probably the Euler-Maclaurin summation formula, which estimates the sum /(O) + /(I) + • • • + f(ri) by the integral j£ f(t) dt, with a correction term expressible as a finite series involving the values of f(t) and its odd derivatives at t = 0 and t = n, together with an explicit error term.

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