I want to compute the following integral numerically in Mathematica, $$\int_{20}^{\infty} \frac{4^{1+4ix}\Gamma(-4ix)e^{-2ix}}{(-2i)^{-4ix}}dx$$ The problem is that when evaluate the integral from $20$ to like $200$, Mathematica gives
Catastrophic loss of precision in the global error estimate due to insufficient
WorkingPrecisionor divergent integral.
but when I break the integral into several pieces and integrate numerically over intervals $$(20,25),(25,35)+\dots+(190,200)$$ and add them up, I get a finite answer. It is not possible to break the integral to several pieces all the way to infinity so I want to numerically integrate this integral from $20$ to infinity without breaking that apart. Could you please help me with this? Thanks!
As @KStarGamer commented, there is no way to say if the integral will converge or not.
Hoping to make life easier, I simplified the integrand $$f(x)=\frac{4^{1+4ix}\Gamma(-4ix)e^{-2ix}}{(-2i)^{-4ix}}=4^{1+6 i x} e^{2 (\pi -i) x} \Gamma (-4 i x)=$$ $$4 e^{2 \pi x} \cos \big[2(6 \log (2)-1)x\big]\,\Gamma (-4 i x)+ i\,\, 4 e^{2 \pi x} \sin\big[2(6 \log (2)-1)x\big]\,\Gamma (-4 i x)$$ which did not make any difference with any of the CAS or integration tools I have been able to use.
Doing the same as you did, that is to say computing numerically $$S_p=\sum_{k=2}^p \int_{10 k}^{10 (k+1)} f(x) \, dx$$ Look how chaotic are the partial sums $$\left( \begin{array}{ccc} p & \Re(S_p) & \Im(S_p) \\ 2 & 0.108953 & +0.022820 \\ 3 & 0.113746 & -0.096633 \\ 4 & 0.112116 & -0.087309 \\ 5 & 0.050178 & -0.060335 \\ 6 & 0.096228 & -0.009431 \\ 7 & 0.121693 & -0.052570 \\ 8 & 0.115593 & -0.060880 \\ 9 & 0.104341 & -0.021764 \\ 10 & 0.091119 & -0.072073 \\ 20 & 0.105419 & -0.042355 \\ 30 & 0.086591 & -0.058269 \\ 40 & 0.078336 & -0.045741 \\ 50 & 0.081443 & -0.050922 \\ 60 & 0.094577 & -0.039728 \\ 70 & 0.092210 & -0.039019 \\ 80 & 0.089834 & -0.038934 \\ 90 & 0.083845 & -0.042020 \\ 100 & 0.083572 & -0.047953 \\ 200 & 0.085813 & -0.047719 \end{array} \right)$$