The only question I see is "do you understand the difference between an equation and an identity?" After looking up the difference, I do not understand complex analysis or the gamma function any better, so I chose to refrain from commenting on an element of the discussion that would lead it in an unproductive direction. Otherwise, what other questions did you have?

At this point I think I'm just missing is inverse trig identities with complex arguments, the rest is just practice with using those trig identities in equations.

It may assist you better if you let StudioT know where you at mathematically because he's pretty good.

3 semesters of calculus and one of statistics. Though, I see this discussion being wrapped up sooner rather than later, it seems trigonometric functions with complex arguments have rather straight forward identities. Besides inverse trig complex argument identities, the rest are details, perhaps for another thread if they are discussed more in depth.