Non-numerical values [closed]

Question

I want to numerically integrate

l[t_?NumericQ]:=(Cos[-3.2 kx]Cos[-0.999957 t * kz])/(0.0000859982 kz^2+kx^2+ky^2);

NIntegrate[l[t],{kx,-\[Infinity],\[Infinity]},{ky,-\[Infinity],\[Infinity]},{kz,-\[Infinity],\[Infinity]}]

but get the following error

NIntegrate::inumr: The integrand l[t] has evaluated to non-numerical values for all sampling points in the region with boundaries {{[Infinity],0.},{[Infinity],0.},{[Infinity],0.}}.

As you can see I've already tried the ?NumericQ method, but it didn't change anything, the error appears either way. What can I do?

When I change the integration boundaries to avoid the ones that cause a problem according to the error message I still get the same warning but with the changed values, which I also don't understand.

Answer 1

You have a multidimensional integral with a highly-oscillating integrand. This makes a problem since the Methods for integrating highly-oscillating functions work better for 1D integrals. In your case, one can "help" Mma to solve this integral by integrating it over x and y first. I will also replace 0.999957 by 1, while 3.2 and 0.0000859982 - by Rationalize[3.2] and Rationalize[0.00009]:

  int = Assuming[{z \[Element] Reals && z != 0}, 
  Integrate[(Cos[-Rationalize[3.2] x] Cos[-t*z])/(Rationalize[
        0.00009]*z^2 + x^2 + 
      y^2), {x, -\[Infinity], \[Infinity]}, {y, -\[Infinity], \
\[Infinity]}]]

(*  2 \[Pi] BesselK[0, 6/125 Sqrt[2/5] Abs[z]] Cos[t z] *)

Now let us define the following function:

f[t_] := NIntegrate[
   2 \[Pi] BesselK[0, 6/125 Sqrt[2/5] Abs[z]] Cos[
     t z], {z, -\[Infinity], \[Infinity]}, Method -> "LevinRule"];

and make a table:

lst = ParallelTable[{t, f[t]}, {t, -0.2, 0.2, 0.005}];

yielding this:

ListPlot[lst]

Have fun!