Disguising a complex function as a real function.

Question

I had an idea I was wondering how I would go about it. I have only just started high school calculus so forgive my inexperience. Imagine the 3-dimensional plot of a complex function, $F(a+bi)$. There is the real dimension, the complex dimension, and the function's dimension. Now is there a way to look at this function only in the real dimension and the $F(a+bi)$ dimension, yet plotting all of the complex values onto the 2-D surface? The only way I can describe it is like this. Imagine looking straight at the real number line of this 3-D graph and removing the depth of the complex dimension so that you're left with a 2-D plane but keeping the functional plots of all the complex values. How can we manipulate the function itself to not be changed, but remove this dimension and stick all the complex values onto the real plane. If anyone can understand what I'm trying to say, could they rephrase it in math lingo and help show me how to mathematically accomplish this? Thanks.

Answer 1

You want to project the graph of $z = F(x+iy)$ onto the plane $y = 0$.

The problem with doing this is the graph of $z = F(x+iy)$ is an undulating surface. When you project in onto the $xz$-plane, it will in general fill large portions of this plane, and for functions of particular interest, quite often fills the entire plane. So instead of getting something useful that helps to you in understanding the function's behavior, you just get a big blob that doesn't show you much at all about the function.

You can improve on this a little by using color - assigning a sprectrum to points based on the value of $y$, but this only works when each point in the final graph comes from exactly one point of the function (i.e. you never have $F(x + iy_1) = F(x + iy_2)$ for some $x, y_1, y_2$). Another alternative is to not to to project the points for all $y$, but only for some sample, such as $y = 0, 1, 2, ...$. This provides a series of curves that in some cases may be useful in visualizing the actual function behavior. In others, though, it just provides a jumbled mess.

Generally, your best bet is a true 3D graph.