8 replaced http://mathematica.stackexchange.com/ with https://mathematica.stackexchange.com/

Here I introduced a minushankel function because of another issue, I've started a separate posta separate post for it.

Here I introduced a minushankel function because of another issue, I've started a separate post for it.

Here I introduced a minushankel function because of another issue, I've started a separate post for it.

Notice removed Draw attention by xzczd
Bounty Ended with bbgodfrey's answer chosen by xzczd
7 edited tags
6 1. add the missing 1/(2 Pi I) coefficient of inverse Laplace transform. 2. Remove the incorrect description of the integral. (What's highly oscillatory is BesselJ[0, ξ] f[p, ξ], not f[p, ξ] )

# Numerical inverse Laplace-Hankel transform for a highly oscillatory function

When trying to reproduce the result of this paper about numerical solution of Lamb's problem, I encountered the following double integral (to be more precise, the 0-order inverse Hankel-Laplace transform) of a highly oscillatory function $$f(p, \xi)$$:

$$\int _{\gamma -i \infty }^{\gamma +i \infty }e^{pt}\int _0^{\infty }f(p,\xi )\xi J_0(\xi r)d\xi dp$$$$\frac{1}{2 \pi i} \int _{\gamma -i \infty }^{\gamma +i \infty }e^{pt}\int _0^{\infty }f(p,\xi )\xi J_0(\xi r)d\xi dp$$

With[{t = 2},
AbsoluteTiming[
1/(2 Pi I) NIntegrate[ξ BesselJ[0, ξ] f[p, ξ] Exp[p t], {p, -I ∞,
I ∞},
{ξ, 0, ∞}, WorkingPrecision -> 16, MaxRecursion -> 40,
Method -> "ExtrapolatingOscillatory", Exclusions -> Denominator@f[p, ξ] == 0]]]
(* Returned unevaluated after some warning *)


# Numerical inverse Laplace-Hankel transform for a highly oscillatory function

When trying to reproduce the result of this paper about numerical solution of Lamb's problem, I encountered the double integral (to be more precise, the 0-order inverse Hankel-Laplace transform) of a highly oscillatory function $$f(p, \xi)$$:

$$\int _{\gamma -i \infty }^{\gamma +i \infty }e^{pt}\int _0^{\infty }f(p,\xi )\xi J_0(\xi r)d\xi dp$$

With[{t = 2},
AbsoluteTiming[
NIntegrate[ξ BesselJ[0, ξ] f[p, ξ] Exp[p t], {p, -I ∞,
I ∞}, {ξ, 0, ∞}, WorkingPrecision -> 16, MaxRecursion -> 40,
Method -> "ExtrapolatingOscillatory", Exclusions -> Denominator@f[p, ξ] == 0]]]
(* Returned unevaluated after some warning *)


# Numerical inverse Laplace-Hankel transform

When trying to reproduce the result of this paper about numerical solution of Lamb's problem, I encountered the following double integral (to be more precise, the 0-order inverse Hankel-Laplace transform):

$$\frac{1}{2 \pi i} \int _{\gamma -i \infty }^{\gamma +i \infty }e^{pt}\int _0^{\infty }f(p,\xi )\xi J_0(\xi r)d\xi dp$$

With[{t = 2},
AbsoluteTiming[
1/(2 Pi I) NIntegrate[ξ BesselJ[0, ξ] f[p, ξ] Exp[p t], {p, -I ∞, I ∞},
{ξ, 0, ∞}, WorkingPrecision -> 16, MaxRecursion -> 40,
Method -> "ExtrapolatingOscillatory", Exclusions -> Denominator@f[p, ξ] == 0]]]
(* Returned unevaluated after some warning *)

5 Grammar mistake fixed.
4 added 42 characters in body
Notice added Draw attention by xzczd
Bounty Started worth 500 reputation by xzczd