During a conversation I had yesterday, a math professor I occasionally meet with mentioned in passing, "and you might want to try to graph [tex]f(z)=e^z[/tex] on the complex plane...hm...yes...anyway..." (where "z" is complex). So I sat down at Taco Bell yesterday to think about it, and, for a few minutes today at home on my couch, I gesticulated wildly, bending the coordinate grid as best I could without paper. I came up with this: the imaginary axis bends into the unit circle; all lines with constant real values greater than 0 (vertical lines) bend into circles larger than the unit circle; all vertical lines with real values less than 0 bend into circles smaller than the unit circle; all horizontal lines become rays that pass through the angle that corresponds to their imaginary part and almost, but don't quite, touch the origin. Now, my question is, how accurate is my visualization? I don't want to draw it for fear that anything I draw will become too cluttered, but I also don't want to be walking around with the wrong picture of this function in my head. Can anyone either confirm or correct me as necessary? Also, I know there are programs out there that do this sort of thing for you; could anyone direct me to an image of this graph? And, also, one last question: I don't know how to ask this properly, but is there any way to make a representation of the "density" of the values of this function? Because in my head I have the entire negative real axis compressed into values that take up a finite area while the entire positive axis gets free room to roam. Maybe it's a meaningless thought, but, just for visualization purposes, is there a way to add a "density" axis that runs perpendicular to the complex value-plane? Personally, I think it would be pretty. If that makes any sense whatsoever.