I've been playing around with this equation:
$\displaystyle\int_{-\pi}^\pi{\displaystyle\frac{1-e^{3it}}{1-e^{it}}dt}$
Now it seems to me that we can (possibly) split the integral into four seperate integrals; one for each quadrant of the complex plane. Doing this, we can determine the complex part if we know the real part, and vice versa. So I therefore ponder that it might be possible to simplify the exponential expressions into expressions of variables. In other words, I wonder if we can rewrite the $e^{it}$s as $u$s.
I'm extremely interested in the potential of a variable substitution, but maybe I'm missing the obvious. I can't tell you exactly what I'm looking for, except for information concerning the possibility of this method. What is known? If there is a known way, I'd really enjoy seeing it. It would make many series expansions that I'm thinking about much easier, I'm almost certain. If no methods are known, I'd appreciate it very much if someone could take the time to elaborate on all of the reasons why this method is not known.
I'm very interested in going into detail on this matter, because it seems to decide issues concerning a great amount of potential.