Complex "Integrals" of Complex Exponential Planes
More details, some corrections
These only look at angle changes, they are not the real integrals because they do not include the ratio at which "r" increases. So all I'm looking at is a pattern based on the integrals of the ratios obtained through sin and cosine of the "rise over run" (f'(y)*y/[g'(x)*x]) of the equation. Basically treating the differential changes of the tangent of the angle between them as my new angle. This has no direct reason, just that I'm using this tangent as a ratio that corresponds to further changes in the graph strictly based on algebraic changes in x and y. This also means that the graph's are not the true integrals, but represent the overall changes in that specific equation's curve. There's also a weird thing I found that, assuming the equation can ultimately be represented by re^(iz), the derivative of the corresponding equations' "angle ratio" is proportion/congruent to the Integral of that same equation. Without an "r", f'(z)=-F(z), coming from the infinite series, and every 4 terms, the equation is equal to zero. Since the angle ratio will likely be differentiable, it's derivative will impact the both the Integrals, and derivatives, of the complex equation in question. Upon adding these 4, the value should always be a non-zero complex number, which will represent the rotation of your equation. Again though, r will also impact the equation each time, these equations don't "move" to the proper curve, so they can ONLY be thought of as rotations around an origin, no translation should take place. With a proper integral, the resulting equation will be a super set containing every second curve of its derivative.
These graphs are special in that they are technically derivatives, but are stretched out from the integration, allowing you to see how and where each curve moved to. They are a bit messy, so you may need to zoom out extremely far, or close in to separate the lines out or find the pattern. Also, the equation for the "integrals" was obtained by using a few small integrals of (x+yi)^n (n=1,2,3), then simplifying them into a summation based on their patterns (was actually the difference of two very similar summations), then finally turning the resulting expression into a complex equation (in that equation, x was actually y/x, but variables on this site are limited so x was used again) based on dy/dx. Lastly, not all curves appear to be represented in the integrals. My reasoning for why this is the case is that not every part rotates, or a bunch of things rotate in their entirety before every curve has "rotated", meaning their end value would equal the initial resulting in their values being subtracted (abstractly) from the integral. The integral is then composed of all the curves that only rotate a single time every rotation. This being an infinite sum, I would also assume that with the higher the power, the more you would have to "add" before anything gets "subtracted", but this is only because I want to explain why n=[5,10] are freakishly large compared to their equations (all integrals are larger though).
If you can't tell, this is all self study, it's more of an experiment, and it can be applied properly to some degree of success, so that's why I'm calling them integrals. The complex math behind this is also something loosely based on classical techniques (arc integration, W function), but derived through geometric interpretation and failed attempts. So it likely will look like garbage to many of you, but I swear there is a significant amount of rigor behind what I'm doing.