The NIntegrate cannot give a correct answer by Mathematica

Question

Here I want to compute this integral using the Mathematica $$\int_0^\infty \int_{-\pi}^{\pi}s^{\frac{1}{2}+2} \exp (-s) \cos \left(\frac{f}{2}\right) \exp (-i k s) \exp \left(-\frac{\sqrt{8 s u}}{\cos \left(\frac{f}{2}\right)}\right)\; df\, ds$$

For simplicity, I use $w$ as the upper limit of the integral with respect to $s$.

Thus, we have the following code by Mathematica:

Gamm[k_, u_] :=2/(15*Sqrt[Pi])*
  NIntegrate[Exp[-I*k*s]*s^(1/2+2)*
    Exp[-s]*Cos[f/2]*Exp[-Sqrt[8*s*u]/Cos[f/2]],
  {f, -Pi, Pi}, {s, 0, w}];
Table[Gamm[0, u], {u, 0, 0.1, 0.001}]

IF I choose w = 10, Mathematica gives:

Table[Gamm[0, u], {u, 0, 0.1, 0.001}]
(*  {0.99443, 0.789459, 0.722113, 0.675455, 0.639067, 0.609022,
     0.58335, 0.560903, 0.540948, 0.522983, 0.506647}  *)

When I choose w = 10^6, the above integral yields

Table[Gamm[0, u], {u, 0, 0.1, 0.001}]
(*  {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}  *)

Clearly, the integral is not zero. How to solve this problem? It seems that there is something wrong.

Any suggestion is helpful! Thanks!

Answer 1

The following problem is hidden by NIntegrate, but which is one of the common sources of the NIntegrate::izero message you get from the OP's code.

Exp[-I*k*s]*s^(1/2 + 2)*Exp[-s]*Cos[f/2]*
   Exp[-Sqrt[8*s*u]/Cos[f/2]] /.
 {k -> 0, u -> 0.1} /. {f -> 1., s -> 1000.}

General::munfl: Exp[-1032.23] is too small to represent as a normalized machine number; precision may be lost.

(*  0.  *)

Tip: You should make some sort of exploration of the integrand when a numerical integral misbehaves. The two first things would be to evaluate it at a typical point in the integration region and to plot it (if possible, i.e., if it's a single- or two-variable function).

Gamm[k_, u_] := 
  2/(15*Sqrt[Pi])*
   NIntegrate[
    Exp[-I*k*s]*s^(1/2 + 2)*Exp[-s]*Cos[f/2]*
     Exp[-Sqrt[8*s*u]/Cos[f/2]], {f, -Pi, Pi}, {s, 0, w}, 
    PrecisionGoal -> 8, WorkingPrecision -> 32];

Block[{w = 10^6},
 Table[Gamm[0, u], {u, 0, 1/10, 1/100}]
 ]

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

NIntegrate::slwcon: ....

(*
  {1.0000000000626411805661509309408, 
   0.50827294088807703485723248346197, 
   ...
   0.16444683406465457205578964022320, 
   0.15112873347891652231157354357054}
*)

The message is just a warning.

I'm not sure why the integral in TeX in the OP is not the integral being coded:

Block[{w = Infinity},
 Table[Gamm[0, u], {u, 0, 1/10, 1/100}]
 ]
(*
  {0.99999999998679967008116173695908,
   0.50827294069524647486336322647360, 
   ...
   0.16444683409559630424360181571660,
   0.15112873348047743265847037936301}
*)