A novel approach for the determination of asymptotic expansions of certain oscillatory integrals

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

“…On many occasions, it is desirable to obtain an asymptotic estimate of the integral for arbitrarily large b [1,5]. It is well known that, if f (x) = Φ(x) is infinitely differentiable at the origin, the Hankel integral has the Poincaré asymptotic expansion (PAE) [6,[8][9][10][11] 2 g(x 2 )xJ 0 (ωx) dx which arises from high-energy nuclear physics [6,[12][13][14][15].…”

“…This may happen, for example, when the integral is exponentially small in b, so that it is beyond all orders and equation (1.1) reduces to ∞ 0 Φ(x)J ν (bx) dx ∼ 0 + 0 + 0 + • • • as b → ∞. A well-known example of this case is the class of integrals I(ω) = ∞ 0 e −x 2 g(x 2 )xJ 0 (ωx) dx which arises from high-energy nuclear physics [6,[12][13][14][15]. Also for integer values of ν, equation (1.1) terminates, giving the impression that the integral reduces to a polynomial in inverse power of b in the asymptotic limit.…”

“…But, when f (z) is an entire function, they obtained only an order estimate, so that no explicit asymptotic expansion was available. Gabutti & Lepora [15] later tackled the same integral and its generalization given by the integral ∞ 0 f (x 2 )J ν (ω)x ν+1 dx. Their method, which is based on the Laplace transform, gives explicit asymptotic expansions for a limited class of exponentially small Hankel integrals.…”