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I am trying to numerically integrate the following double integral in MATHEMATICA. This question has also been asked at Wolfram Community

expression

where $Im$ is the imaginary part of the expression, $i$ is the imaginary number, $x$ and $y$ are variables while $a, b, c$ and $Q$ are constants greater than 0.

Here is my attempt to solve this.

a = 3
b = 0.0137
c = 0.0023
Q = 6
NIntegrate[
  Im[Exp[-I x c + b (I x y/(y^a - I x))]]/x, {x, 0, ∞}, {y, 0, Q},
  AccuracyGoal -> 10]

Is this the correct way of applying numerical integration with more than one variable? I am getting an error when I run this expression which reads as

evaluated to non-numerical values for all sampling points in the \region with boundaries {{[Infinity], 0.},{0, 6}}.

Can anyone please guide me to correct the implementation of the expression given above.

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    $\begingroup$ Your integrand should be Im[Exp[-I x c + b (I x y/(y^a - I x))]]/x not Im[Exp[-I x c + b (i x y/(y^a - I x))]]/x. Voting to close. $\endgroup$
    – Anton Antonov
    Commented Sep 4, 2020 at 15:02
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    $\begingroup$ This integral does not converge because for small x the integrand behaves as 1/x. $\endgroup$
    – Artes
    Commented Sep 4, 2020 at 15:10
  • $\begingroup$ @Artes Is there any way around to avoid that? If the x values are restricted to be a larger value then it should work, right? $\endgroup$
    – shahrukh
    Commented Sep 4, 2020 at 15:37
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    $\begingroup$ When you simultaneously cross-post the same question (community.wolfram.com/groups/-/m/t/2070465), you should mention that in both posts. $\endgroup$
    – JimB
    Commented Sep 4, 2020 at 16:17
  • $\begingroup$ @Artes: The integral under consideration converges at the origin in view of ComplexExpand[ Normal[Series[ Im[Exp[-I x c + b (I x y/(y^a - I x))]]/x , {x, 0, 1}]]] /. {x -> r*Cos[\[Phi]], y -> r*Sin[\[Phi]]}] which results in -(1/r)E^(-((137 r^3 Cos[\[Phi]]^2 Sin[\[Phi]])/( 10000 (r^2 Cos[\[Phi]]^2 + r^6 Sin[\[Phi]]^6)))) Sec[\[Phi]] Sin[(23 r Cos[\[Phi]])/10000 - ( 137 r^5 Cos[\[Phi]] Sin[\[Phi]]^4)/( 10000 (r^2 Cos[\[Phi]]^2 + r^6 Sin[\[Phi]]^6))].. $\endgroup$
    – user64494
    Commented Sep 5, 2020 at 6:45

1 Answer 1

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ComplexExpand the functions and integrate separately.(Because I am in a hurry, i don't show intermediate results)

ceRe = ComplexExpand[Re[(I x y)/(y^3 - I x)], 
          TargetFunctions -> {Re, Im}]

ceIm = ComplexExpand[Im[(I x y)/(y^3 - I x)], 
          TargetFunctions -> {Re, Im}]

intRe[x_] = Integrate[ceRe, {y, 0, 6}, Assumptions -> x > 0]

intIm[x_] = Integrate[ceIm, {y, 0, 6}, Assumptions -> x > 0]

ii[x_] = ComplexExpand[Im[Exp[-I x c + b*(intRe[x] + I intIm[x])]]/x, 
TargetFunctions -> {Re, Im}] /. {b -> 137/10000, c -> 23/10000} //
   Simplify[#, x > 0] &

Plot[ii[x], {x, 0, 10000}]

nint = NIntegrate[ii[x], {x, 0, \[Infinity]}, MaxRecursion -> 50]

(*   -0.989098   *)
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