can anyone explain what the $\arg\Gamma (ix)$ is?
I am largely unclear on what the gamma function is also and how it is defined for complex numbers.
I know how the argument of a function is normally defined, but I am unclear on this gamma function so can't really decide what to do. I have tried a google search and looked at the wiki page but am getting know where really.
$$\Gamma(z) = \int_0^{\infty} dt \: t^{z-1} e^{-t}$$
$$\Gamma(i x) = \int_0^{\infty} dt \: t^{i x-1} e^{-t} = \int_0^{\infty} \frac{dt}{t} \: e^{i x \log{t}-t}$$
Expand into sines and cosines
$$\Gamma(i x) = \int_0^{\infty} \frac{dt}{t} \: e^{-t} [\cos{(x \log{t})} + i \sin{(x \log{t})}]$$
From this, in principle, $\arg{\Gamma(i x)}$ may be computed. I would worry about the potential divergence of the integrals near $t=0$.