Numerically Inverting Characteristic Function with Inverse Fourier Transform

Question

I'm trying to numerically invert this characteristic function:

cf = (((-I)*t + n*(λ + μ) - 
       n*Sqrt[-4*λ*μ + (I*t - n*(λ + μ))^2/
           n^2])/(n*Sqrt[λ*μ*ρ]))^n/2^n

via a numerical Inverse Fourier Transform that looks like this...

Re[(1/(2*Pi))*
  NIntegrate[
   cf /. {λ -> 0.5, μ -> 1, ρ -> 0.5, 
     n -> 5}, {t, -Infinity, 0, Infinity}, 
   Method -> DoubleExponential]]

but I keep on getting errors that the integral cannot be evaluated at the boundaries.

Does anyone have recommendations on how to compute this thing so that I get numerical values?

Thanks for your assistance.

Answer 1

The Integrate works for this function:

cf = (((-I)*t + n*(λ + μ) - 
        n*Sqrt[-4*λ*μ + (I*t - n*(λ + μ))^2/
            n^2])/(n*Sqrt[λ*μ*ρ]))^n/2^n;

Re[(1/(2*Pi))*
  Integrate[
   cf /. {λ -> 0.5, μ -> 1, ρ -> 0.5, 
     n -> 5}, {t, -Infinity, Infinity}]]
(* 3.90799*10^-13 *)