Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales

Abstract

We obtain an exactification of the Poincaré asymptotic expansion (PAE) of the Hankel integral,We find that, for halfinteger orders of the Bessel function, the exactified asymptotic series terminates, so that it gives an exact finite sum representation of the Hankel integral. For other orders, the asymptotic series does not terminate and is generally divergent, but is amenable to superasymptotic summation, i.e. by optimal truncation. For specific examples, we compare the accuracy of the optimally truncated asymptotic series owing to the McClure-Wong distributional method with owing to the Mellin-Barnes integral method. We find that the former is spectacularly more accurate than the latter, by, in some cases, more than 70 orders of magnitude for the same moderate value of b. Moreover, the exactification can lead to a resummation of the PAE when it is exact, with the resummed Poincaré series exhibiting again the same spectacular accuracy. More importantly, the distributional method may yield meaningful resummations that involve scales that are not asymptotic sequences.

Direct Measurement of the In-Plane Thermal Diffusivity of Semitransparent Thin Films by Lock-In Thermography: An Extension of the Slopes Method

“…Consider a Hankel integral of zero orderThis is an asymptotic expansion that when b → ∞ can be expressed as 12 where the first term (infinite summation) is the Poincaré asymptotic expansion (PAE) of the integral in eq 4a and is a series expansion containing integer powers of 1/ b . ϕ s (0) represents the s th derivative of ϕ( x ) evaluated at x = 0, and Γ(·) is the gamma function.…”

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. I. Integer orders

“…Not only that the contour integral representation allows unique identification of the divergent integrals as finite part integrals, it is responsible in picking up the poles and branch points of the integrand f (z)(ω + z) −n that are the origins of the missing terms in the naive application of term by term integration of the expansion (11). The missing terms are precisely the residue terms in equations (19) and (20).…”

“…, ν = 0 and entire f (z), the Stieltjes transform will have to take the representation given by equation (19); this is implemented in Section-4. On the other hand, for integral orders λ = n, ν = 0 and entire f (z), the Stieltjes transform will assume the representation given by equation (20); this is implemented in Section-5. For non-integral order λ = n, a contour integral representation other than equations ( 19) and ( 20) will have to be devised in [22].…”

“…The missing terms are recovered by interpreting the divergent integrals as functionals over some fundamental space of test functions so that the singular factors in the integrand are interpreted as distributions. However, despite of the absence of a counter example to the results of McClure and Wong (in fact we have recently utilized their distributional approach to exactify the Hankel transform [20]), it is desirable to clarify the intervening steps leading to their final results. In the intervening steps, divergent integrals arise and they are assigned values by analytic continuation.…”

Asymptotic analysis--a brief survey

“…广义函数法已应用于单侧 Hilbert 变换在无穷远点的渐近展开, 以及 Fourier 变换和 Laplace 变换在 0 点的渐近展开等, 可参见文献 [4, 第 6 章] 和 [17]. 最近, Galapon 和 Martinez [20] 应用此方法, 研究了…”