# Asymptotic Approximation of Integrals by R. Wong

By R. Wong

Asymptotic tools are usually utilized in many branches of either natural and utilized arithmetic, and this vintage textual content continues to be the main up to date e-book facing one vital point of this zone, specifically, asymptotic approximations of integrals. during this publication, all effects are proved carefully, and plenty of of the approximation formulation are observed by means of 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 other places. lots of the examples during this textual content come from concrete functions. due to the fact its e-book twelve years in the past, major advancements have happened within the normal concept of asymptotic expansions, together with smoothing of the Stokes phenomenon, uniform exponentially more suitable asymptotic expansions, and hyperasymptotics. those new techniques belong to the world referred to now as 'exponential asymptotics'. Expositions of those new theories come in papers released in numerous journals, yet no longer but in e-book shape.

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Extra resources for Asymptotic Approximation of Integrals

Sample text

26) give These results couple together to give uniformly in arg z, as z -> oo in 24 I Fundamental Concepts of Asymptotics The above analysis again gives only an asymptotic expansion without an error bound. Although it is possible to modify the argument so that a numerical bound can be constructed for the error associated with the above expansion (see for instance, Olver (1974a, p. 114)), we shall not pursue this possibility, and prefer to illustrate it with some specific examples. Example 4. The Bessel Function Jv(x).

Also, we believe it is a wrong attitude to prescribe an asymptotic sequence {(pn} on an a priori basis and then investigate procedures for producing asymptotic expansions J] /„ for a specific function with respect to {(pn}. 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.

The general properties of Laplace integrals (Widder 1941, p. 37) ensure that F(z) exists as long as Then, by hypothesis, \g(t)\ 2|z0| esc A, where 6 = \ sin A. Hence as z -» oo in \&rg(zel7)\ < n/2 — A. This, of course, implies uniformly in arg z, as z -> oo in |arg(ze£r)l <; rc/2 — A.