Asymptotic estimation of Hankel transforms and the screened Coulomb potential in trilayer structures

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

“…The Hankel integral, ∞ 0 f (x)J ν (bx) dx, arises naturally in many fields of application in physics [1][2][3][4][5] example, which has served as the motivation behind this work, arises in the quantum tunnelling problem where the traversal time across a potential barrier appears in the form of a Hankel integral of the zeroth order [5]. 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]…”

“…With the substitution λ = s in the right-hand side of equation (4.3), we obtain the values of I (1) s (b) and I (2) s (b). Substituting these integrals back into equation (4.1), we arrive at…”

“…To obtain the asymptotic expansion of I 1 (b) for large b, let us instead obtain the asymptotic expansion of the integralĨ 1 for Re(λ) > 0 and Re(β) > 0, where D ρ (z) is the parabolic cylinder function (PCF). The integrals…”

“…In order to obtain an expansion in the form of equation (2.1), we write the Bessel function in terms of the Hankel functions using the relationship 2J ν (z) = H (1) [36]. The Hankel integral (1.2) then splits into two integrals,…”

“…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].…”