Complex Iteration   Ive discovered that not only fractional iteration is possible but also complex iteration Its not as intuitive as fractional iteration but it does make sense once you have a new model for what iteration means Probably the simplest explanation is with orbitals in the complex plain iterating to a real values generates the complex orbitals as paths Iterating to imaginary powers meanwhile generates paths perpendicular to the orbitals at every point You need the entire vector field of orbitals to know the new path For example iterating a linear function multiplying by a constant creates orbitals which are rays Iterating to imaginary powers creates paths with are concentric circles After that complex iteration can be done by composing real and imaginary values For a more precise derivation I had to invent a new type of derivative it looks at the instantaneous change in value at a point as the function iterates The formula for this derivative is  fx  lim n fnx  xn Which is equivalent to the derivative of fs Abel function The key about this derivative is it has the property that fn  f x n  This enables iteration to imaginary powers to be calculated directly Ultimately I think it is equivalent to taking the analytic extension of the Abel function but its a bit more intuitive than Taylor series witchcraft Ive done the calculations to arrive at Eulers identity using this method rather than the typical one and graphically it makes much more sense once its understood how complex iteration works its just geometry to find the imaginary value e must be raised to in order to complete the arc from  to