ElementaryFunctions/Exp/06/GraphingOnTheComplexPlane

[Saint Medici]

During a conversation I had yesterday, a math professor I occasionally meet with mentioned in passing, "and you might want to try to graph $$f(z)=e^z$$ 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.

 

[Integral]

The most common plot of the complex plane is to express the complex number as z=x+iy, Then plot x and y on the Cartesian plane.

 

[Muzza]

You might find http://www.mai.liu.se/~halun/complex/domain_coloring-unicode.html interesting.

 

[Galileo]

Saint Medici,

Your vision is correct. (Although instead of saying vertical lines are 'bend' into circles, I'd say they are 'wrapped around' into circles).
Notice that the function is not injective on C,
but it is injective on a horizontal strip (couldn't get brackets to work for set notation):
$$S_a=\left(z:a<Im(z)\leq a +2\pi\right)$$
This entire strip gets mapped onto the complex plane (without the origin).

Now suppose we take a=0.
So we have that part of the plane with $0<Im(z)\leq 2\pi$.
Try to imagine how this strip gets deformed into a plane with a hole in it.

A vertical line in this strip goes to a circle on the plane. So take the bottom part
and tape it to to the upper part (think of it as a piece of paper or rubber which you can roll so you get a tube).
then you have a cylinder. Now squeeze the left side of the cylinder and stretch the
right side (from the inside) so you get the plane with a hole in the middle.

What I've just done is actually Topology (great fun!) and in mathematical terms it says that the strip is homeomorph with the plane*. (homeomorph here means that the strip and the plane (without the orgin) can be mapped into each other by a continuous invertible function with a continuous inverse). Geometrically, the function is stretching and bending the domain into the codomain.

One of my math teachers said topology is rubbermath. We may bend and stretch, but not cut or tear.

Okay, I've digressed, but topology is always a great way to visualise what functions do!

*EDIT: I noticed an error in my post. The regions are not homeomorphic, because it's inverse (which would be the logarithm) is not continuous. You have to 'cut' the plane to get the strip back. But the construction would still work...

 

[pnaj]

Muzza, that is a really nice link.
Thanks,
Paul.

 

[Zurtex]

Although I am sure working it out yourself is probably best, here is a nice site to take a gander at from time to time: http://functions.wolfram.com/