Complex exponential integrand and NIntegrate

Question

For some non-negative numbers $m_1$, $m_2$, $E$ I define a function $$ f\left(\boldsymbol{q},E\right)=\frac{1}{2\omega_{1}\omega_{2}}\frac{1}{\omega_{1}+\omega_{2}+E}+\frac{1}{4\omega_{1}\omega_{2}}\frac{1}{\omega_{1}-\omega_{2}-E}+\frac{1}{4\omega_{1}\omega_{2}}\frac{1}{\omega_{2}-\omega_{1}-E} $$ where $$ \omega_{1,2}\left(\boldsymbol{q}\right)=\sqrt{m_{1,2}^{2}+\boldsymbol{q}^{2}} $$ Note that $f$ actually depends on $\boldsymbol{q}^2$.

I want to evaluate at least numerically the integral over all $\mathbb{R}^3$ $$ I\left(E\right)=\int\frac{d^{3}q}{\left(2\pi\right)^{3}}f\left(\boldsymbol{q},E\right)e^{i\boldsymbol{q}\cdot\boldsymbol{n}} $$ where $\boldsymbol{n}$ in a non-negative integer 3-vector.

I've tried NIntegrate using e.g. $m_1=0.2$, $m_2=0.7$, $E=0.8$ and $\boldsymbol{n}=\left(1,1,1\right)$ . The calculation takes very long and I just abort. I've tried changing the input numbers, adding some methods as options. At most, I get warnings like NIntegrate::slwcon. I believe Mathematica runs into trouble because of the oscillating exponential.

With a simpler version, where $f$ is just $$ f\left(\boldsymbol{q}\right)=\frac{1}{2\sqrt{m^{2}+\boldsymbol{q}^{2}}} $$ similar issues appear. When it works, after long time, it's quite different from the analytical result.

Are you aware of some magical methods/options combination that would speed up the process and give a reliable result? Of course, if you know some analytical converging expression for the integral, I'd be happy to know.

Answer 1

It looks like the best approach is to use spherical coordinates:

ω1 = Sqrt[m1 + r^2];
ω2 = Sqrt[m2 + r^2];
f = 1/(ω1 ω2) 1/(ω1 + ω2 + e) + 
  1/(2 ω1 ω2) 1/(ω1 - ω2 - e) + 
  1/(2 ω1 ω2) 1/(ω2 - ω1 - e)

(*
==> 1/(
 2 Sqrt[m1 + r^2] Sqrt[
  m2 + r^2] (-e + Sqrt[m1 + r^2] - Sqrt[m2 + r^2])) + 1/(
 2 Sqrt[m1 + r^2] Sqrt[
  m2 + r^2] (-e - Sqrt[m1 + r^2] + Sqrt[m2 + r^2])) + 1/(
 Sqrt[m1 + r^2] Sqrt[m2 + r^2] (e + Sqrt[m1 + r^2] + Sqrt[m2 + r^2]))
*)

Block[{m1 = .2, m2 = .7, e = .8},
 NIntegrate[r^2 f Exp[I z r], {r, 0, Infinity}, {z, -1, 1}]]

(* ==> -1.69435 + 8.38814*10^-11 I *)

Here I chose the direction of $\mathbf{n}$ as the z axis for the integration variable and transformed to spherical coordinates $r, \theta, \phi$. I omitted the factor $2\pi$ coming from the trivial $\phi $ integration and substituted $z=\cos\theta$. The exponential then becomes $\exp(i \mathbf{q}\cdot\mathbf{n}) = \exp(i r \cos\theta)= \exp(i r z)$. The numerical integration is quite fast with the order of integrations chosen as above.