Numerical integration of an integral with singularity

Question

I am trying to solve this integral numerically using Mathematica. Here is my integral

$$\int_0^{\infty} dx\;\frac{\Gamma(\delta-4ix)}{(i(x-1)+\epsilon)^{1-4ix}}\;, $$ where $0<\delta,\ll 1$ and $0<\epsilon\ll 1$. I would like to find this integral as $\delta$ and $\epsilon$ go to zero and I want to do this numerically. I am wondering how I can do that in Mathematica. To clarify my question, I give an example; $$\int_0^{\infty} dx\; \frac{1}{x-1-i\epsilon}=i\pi\delta(x-1)+PV \int_0^{\infty}dx\;( \frac{1}{x-1}) $$ that PV stands for principal value. Using the Cauchy Principal Value method in Mathematica, one can solve this integral numerically. I am thinking if there is a method that can solve my original integral numerically in Mathematica. Please note that this integral may be solved using complex integration, but I am looking for a numerical method. Could you please help me with this? Thanks!

Answer 1

Not sure I completely understand the question, do you want to implement a numerical method to calculate the integral? or you just need a method in Mathematica that already does it?

f[d_, e_] := NIntegrate[Gamma[d - 4 I x] / (I(x - 1) + e)^(1 - 4 d x), {x, 0, Infinity}]

f[0.01, 0.01]

-1.92598 + 0.793114 I