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By R. Wong

ISBN-10: 0898714974

ISBN-13: 9780898714975

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Example text

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.

66). If F(z) has a convergent expansion of the form which can be integrated term by term to give 5. Watson's Lemma 31 then the behavior as t -» 0+ would be established. 71) is not valid. 71). Converse to Watson's Lemma. 68). // 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).

In order to generalize, we consider the integral 22 I Fundamental Concepts of Asymptotics where y is a fixed real number, and the path of integration is the straight line joining t — 0 to t = aoeiy. The following result is given in Wyman (1964, p. 249). Generalized Watson's Lemma. 13) exists for some fixed z = z0, and that as t -> 0 along arg t = y, where Re and Re Then as z -» oo in |arg(zelv)| < rc/2 — A, for any real number A in the interval 0 < A <: n/2. Proof. The general properties of Laplace integrals (Widder 1941, p.

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


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