Double NIntegrate of a double exponential oscillatory function

Question

I have a double integral involving a double exponential oscillatory function:

\begin{equation} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-iq^3} e^{-q^2}(q+b)\text{sign(q-b)}e^{ib^3} e^{-b^2}dq db \end{equation}

i just used the following code

NIntegrate[$e^{-iq^3}$ $e^{-q^2}$ (q+b) $\text{sign(q-b)}$ $e^{ib^3}$ $e^{-b^2}$,{q,-$\infty$,$\infty$},{b,$-\infty$,$\infty$},Method -> {"DoubleExponentialOscillatory", "SymbolicProcessing" -> 0}]

and it gives the following message

Numerical integration converging too slowly; suspect one of the \
following: singularity, value of the integration is 0, highly \
oscillatory integrand, or WorkingPrecision too small. 

I tried using Method -> {"DoubleExponentialOscillatory", "SymbolicProcessing" -> 0} in NIntegrate and it gives the following message

"Method \!\(\"DoubleExponentialOscillatory\"\) works only for \
one-dimensional integrals with infinite ranges. \
\!\(\*ButtonBox[\">>\", Appearance->{Automatic, None}, \
BaseStyle->\"Link\", ButtonData:>\"paclet:ref/NIntegrate\", \
ButtonNote->\"NIntegrate::oscir\"]\)"

How do I use NIntegrate? Thanks in advance

Answer 1

For every point {q, b} in the integrand, there is another point with the values of q and b interchanged, and the value of the integrand is the negative of the first point. So, the integral must be zero, and that is why Mathematica gives the first error message in the question.

This can be seen explicitly as follows. Define variables x and y as follows.

Solve[{q - b == x, q + b == y}, {q, b}] // First

(* {q -> (x + y)/2, b -> 1/2 (-x + y)} *)

Then, the integrand becomes

Simplify[Exp[-q^2 - I q^3 - b^2 - I b^3] (q + b) Sign[q - b] /. %]

(* E^(-(1/4) I (y^2 (-2 I + y) + x^2 (-2 I + 3 y))) y Sign[x] *)

and

Integrate[%, {x, -Infinity, Infinity}]

(* 0 *)

as expected.