Absolutely nothing special, I tried a new idea I had with complex planes, and it sorta worked.
In essence, take the second integral for the area of a circle (in terms of "r" not "x" and "y") and add it to the third multiplied by i. Next divide the imaginary part by the real VARIABLES (leave everything else alone, ONLY the variable), treat that as the sin of some angle, per euler's formula, and derive the expression for the angle (sin^-1). Then just simplify it, and integrate the radical, most things cancel out if this is done correctly so it usually just ends up working itself out when you simplify. Luckily, I don't have an imaginary part in the equation for the area of a circle, so I can graph my equations in the real plane. For the first time in months, this is actually on the real plane.
This is a circle at n=3 because I'm leaving out the real part of this expression in order to "keep it real". There's a striking relationship between the x and y axis as well. X never changes, whereas y shrinks as n increases, forming an ellipse. I'm probably a little to optomistic, but maybe you could work this around some way to find an expression for the perimeter of an ellipse, who knows.
