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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.

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7 edited tags
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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]` )
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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.
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4 added 42 characters in body
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3 Correct mistakes caused by a typo in the paper.
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2 Typo fixed.
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1
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